Besides,
taking away your stress
exercise is proven to increase concentrations of norepinephrine, a chemical that can moderate the brain’s response to stress.
and boosting happy chemicals
usually within five minutes after moderate exercise you get a mood-enhancement effect.
and alleviating long-term depression
data suggests that active people are less depressed than inactive people, those who were active and stopped tend to be more depressed than those who maintain or initiate an exercise program
and improving your self-confidence
regardless of weight, sexual orientation, gender, or age, exercise can quickly elevate a person's perception of his or her attractiveness: self-confidence and self-worth.
and preventing cognitive decline
exercising boosts the chemicals in the brain that support and prevent degeneration of the hippocampus, an important part of the brain for memory and learning, which would prevent Alzheimer.
and reducing anxiety
the warm and fuzzy chemicals that are released after exercise can help people with anxiety disorders calm down
and help controlling addiction
we all are addicted to something, right? Whether it is drugs, alcohol, internet, food or sex, exercise can help in addiction recovery
and unleashing your creativity
It is reported that one gym session can boost creativity for up to two hours afterwards.
and reducing the chances of developing heart disease, stroke, and diabetes
It is reported by over 11 scientific articles that bicycling and walking reduce your chance of getting all those diseases.
and making you lose weight
you don't need evidence for that? do you?
and promoting better sleep
regular physical activity can help you fall asleep faster and sharpen your sleep.
and lowering risk of all the following
up to a 35% lower risk of coronary heart disease and stroke and up to a 50% lower risk of type 2 diabetes and up to a 50% lower risk of colon cancer and up to a 20% lower risk of breast cancer and a 30% lower risk of early death and up to an 83% lower risk of osteoarthritis and up to a 68% lower risk of hip fracture and a 30% lower risk of falls (among older adults) and up to a 30% lower risk of depression and up to a 30% lower risk of dementia
and preventing osteoporosis
Weight-bearing exercises, like running, walking and weight-lifting, help lower your odds of getting osteoporosis as you grow older
and reducing the severity of asthma
swimming is one of the best exercises for people with asthma, exercise helps in fewer or milder asthma attacks overall and a need for less medication.
and promoting a healthy pregnancy
Relaxation exercises and Kegel exercise that strengthen the pelvic muscles and back exercises are all important for pregnant women.
and having anti-ageing effects
exercise might be the best anti-ageing pill according to many scientists.
and improving your sex life
The medical research points towards it: the fitter you are, the better your sex life is.
and ignoring the fact that physical inactivity is twice as deadly as obesity
from recent report by the American Journal of Clinical Nutrition.
there is no evidence that exercise makes you healthier at all.
UPDATE:
The OP updated his question saying he's asking about causation and not correlation.
It seems:
1) there is a lack of evidence that a causal link between physical exercise and good mental health exists.
This book called Workplace Health: Employee Fitness And Exercise provides a great summary to a series of studies which concluded that there is a lack of causal relationship between physical exercise and mental health.

In another large cross-sectional research(1), scientists and researchers were able to prove that physical exercise is correlated with lower risk of depression, however, fortunately for people against physical exercise, they couldn't prove a causal link between the two, researchers said they:
are not able to make any firm conclusions on the direction of causation in any of the associations described
NHS blogged about the study and said:
In general, this study adds to the evidence that exercise is good for mental health, although in isolation it does not prove a causal link between the two. It should be discussed in the context of what else is known about the benefits of exercise.
Digging deeper, I found that, according to a governmental report:
2) there is a causal link between physical exercise and reduction in risk in all-cause mortality, all CVDs combined, CHD, hypertension, colon cancer, and NIDDM.
Causality
The studies reviewed in this chapter indicate that
physical activity is associated with a reduction in risk
of all-cause mortality, all CVDs combined, CHD,
hypertension, colon cancer, and NIDDM.
To evaluate whether the information presented is sufficient to
infer that these associations are causal in nature, it is
useful to review the evidence according to Hill’s
classic criteria for causality (Hill 1965; Paffenbarger
1988).
Strength of Association. The numerous estimated
measures of association for cardiovascular outcomes
presented in this chapter generally fall within the
range of a 1.5- to 2.0-fold increase in risk of adverse
health outcomes associated with inactivity. This
range represents a moderately strong association,
similar in magnitude to the relationship between
CHD and smoking, hypertension, or elevated cholesterol.
The associations with NIDDM, hypertension,
and colon cancer have been somewhat smaller
in magnitude. The difficulty in measuring physical
activity may lead to substantial misclassification,
which in turn would bias studies toward finding less
of an effect of activity than may actually exist. On the
other hand, not controlling for all potential confounders
could bias studies toward finding more of
an effect than may actually exist. Efforts to stratify
studies of physical activity and CHD by the quality of measurement have found that the methodologically
better studies showed larger associations than those
with lower quality scores (Powell et al. 1987; Berlin
and Colditz 1990). In addition, cardiorespiratory
fitness, which is more objectively and precisely measured
than the reported level of physical activity,
often is also more strongly related to CVD and
mortality. Measures of association between physical
activity and health outcomes thus might be stronger if
physical activity measurements were more accurate.
Consistency of Findings. Although the epidemiologic
studies of physical activity have varied greatly
in methodology, in ways of classifying physical activity,
and in populations studied, the findings have
been remarkably consistent in supporting a reduction
in risk as a function of greater amounts of
physical activity, or conversely, an increase in risk as
a function of inactivity.
Temporality. For most of the health conditions
included in this chapter (all-cause mortality, CVD,
CHD, hypertension, NIDDM), longitudinal data from
cohort studies have been available and have confirmed
a temporal sequence in which physical activity
patterns are determined prior to development of
disease. For obesity and mental health, fewer longitudinal
studies have been conducted, and findings
have been more equivocal. Perhaps the strongest
evidence for temporality comes from two studies of
the effect of changes in activity or fitness level. Men
who became more active or more fit had a lower
mortality rate during follow-up than men who remained
inactive or unfit (Paffenbarger et al. 1993;
Blair et al. 1995).
Biological Gradient. Studies of all-cause mortality,
CVD, CHD, and NIDDM have shown a gradient
of greater benefit associated with higher amounts of
physical activity. Most studies that included more
than two categories of amount of physical activity
and were therefore able to evaluate a dose-response
relationship found a gradient of decreasing risk of
disease with increasing amounts of physical activity
(see Tables 4-1 through 4-8).
Biologic Plausibility.
Evidence that physiologic effects of physical activity have beneficial consequences
for CHD, NIDDM, and obesity is abundant
(see Chapter 3, as well as the biologic plausibility
sections of this chapter). Such evidence includes
beneficial effects on physiologic risk factors for
disease, such as high blood pressure and blood
lipoproteins, as well as beneficial effects on circulatory
system functioning, blood-clotting mechanisms,
insulin production and glucose handling,
and caloric balance.
Experimental Evidence. Controlled clinical trials
have not been conducted for the outcomes of mortality,
CVD, cancer, obesity, or NIDDM. However,
randomized clinical trials have determined that
physical activity improves these diseases’ risk factors,
such as blood pressure, lipoprotein profile,
insulin sensitivity, and body fat.
The information reviewed in this chapter shows
that the inverse association between physical activity
and several diseases is moderate in magnitude, consistent
across studies that differed substantially in
methods and populations, and biologically plausible.
A dose-response gradient has been observed in
most studies that examined more than two levels of
activity. For most of the diseases found to be inversely
related to physical activity, the temporal
sequence of exposure preceding disease has been
demonstrated. Although controlled clinical trials
have not been conducted (and are not likely to be
conducted) for morbidity and mortality related to
the diseases of interest, controlled trials have shown
that activity can improve physiologic risk factors for
these diseases.
From this large body of consistent information, it is reasonable to conclude that physical activity is causally related to the health outcomes reported here.
References:
Layman, E. M. (1960). Contribution of exercise and sport to mental health and social adjustment. In Science and medicine of exercise and sports, ed. W.R. Johnson, New York
Cureton, T. K. (1963) Improvement of psychological state by means of exercise programs. Journal of the Association for Physical and mental rehabilitation, 17, 14-25.
Chernen, L., FRIEDMAN, S., GOLDBERG, N., FEIT, A., KWAT, T. and STEIN, R. (1995) Cardiac disease and ...
Morgan, W.P. and O'Connor, P. J. (1988) Exercise and mental health. In Exercise adherence: its impact on public health, ed. R.K. DISHMAN. Champaign.
(1) Harvey SB, Hotopf M et al. Physical activity and common mental disorders. The British Journal of Psychiatry (2010) 197: 357-364