TL;DR Answer: No firm studies have been done that I could find, however: Given the general ineffectiveness of the rhythm method, and the rate of spontaneous abortions, it follows that more fertilized eggs "die" as a result than if an egg was not ever allowed to become fertilized through other barrier or pharmacological methods.
Long Answer with citations:
According to the National Institute of Health (same citation you had now that I look at the link in your question. Other citations: University of Maryland, University of Pittsburgh, Standford University, University of Rochester with WHO data I'll come back to some of these), amongst women who know they are pregnant, the rate is 15-20%. That's the most reliable information we can get, because of the knowledge factor. Otherwise the NIH estimates a 50% spontaneous abortion rate.
Most miscarriages are caused by chromosome problems that make it impossible for the baby to develop. Usually, these problems are unrelated to the mother or father's genes.
Other possible causes for miscarriage include:
Drug and alcohol abuse
Exposure to environmental toxins
Physical problems with the mother's reproductive organs
Problem with the body's immune response
Serious body-wide ( systemic) diseases in the mother (such as uncontrolled diabetes)
It is estimated that up to half of all fertilized eggs die and are lost (aborted) spontaneously, usually before the woman knows she is pregnant. Among those women who know they are pregnant, the miscarriage rate is about 15-20%. Most miscarriages occur during the first 7 weeks of pregnancy. The rate of miscarriage drops after the baby's heart beat is detected.
Given that the rate is 15-20% for those who know, then it can be logically concluded that some women will never know they are pregnant (or have a fertilized egg) and will have a spontaneous abortion (miscarriage).
Now, back to the links I provided earlier that were not NIH:
University of Rochester:
Approximately 10 to 20 percent of clinically recognized pregnancies under 20 weeks of gestation will undergo spontaneous abortion
Stanford University: A paper from 1975 entitled Estimation of Spontaneous Abortion Rates, shows a rate of about 12-26%.
University of Pittsburgh says:
Approximately one in four pregnant women experience one or more miscarriages, making spontaneous abortion the most common pregnancy complication, and of public health importance.
Again, the biggest weakness in all these studies is that they only account for the number of women who knew they were pregnant. The only study I found was from the University of Michigan Medical School that said:
The incidence of clinically unrecognized loss is approximately 20%.
I did find some sites that claimed: "some research has indicated that 70% of fertilized eggs may never go on to become a full-term pregnancy" however, none of them were referenced to a medical study. With the understanding that it's hard to collect data when someone doesn't even know that they are indeed technically pregnant.
Okay, with all this lead in material about how many actual pregnancies actually end in a miscarriage, we need to look at the effectivity of the Rhythm Method. First of all, the better term for this method is "Fertility Awareness" and here the University of Illinois gives some data (note the highlighted portion by me).
Method: Fertility Awareness
Effectiveness: 75 - 97%
STD Protection: No
Utilizes daily body temperature charts, sometimes combined with daily cervical mucus checks to predict ovulation which indicates risk of pregnancy.
Requires instruction, high motivation and diligent record-keeping of fertility indicators.
Increases awareness of changes in menstrual cycle.
Requires use of back-up method or abstinence from intercourse during fertile part of cycle.
Can be an "all natural" method.
Stress, illness or vaginal infection can affect fertility indicators, which can result in pregnancy.
Not the same as "Rhythm Method" which is not a reliable method of birth control.
From the University of Utah medical library, they have information about using fertility awareness. Again, not the highlight I made:
This is also known as "natural family planning" or periodic abstinence. These include (1) the rhythm or calendar method, (2) the basal body temperatures (BBT) method, (3) the mucus method, ovulation or Billings method, and (4) the symptothermal method. They all require couple education and motivation. It is key to remember here that ovulation occurs 14 days +/- two days before menstruation. The follicular phase of the menstrual cycle is the one that varies in length, not the luteal phase. Therefore, a woman can look back and say when she ovulated, but not as easily ahead to predict the time of the next ovulation.
The rhythm method can only be used effectively in a woman with regular cycles. It is based on three assumptions: (1) The ovum can be fertilized for about 24 hours after ovulation. (Often states as up to 72 hours, but this would be a very rare exception.) (2) Spermatozoa retain their fertilizing ability for only about 48 hours. (3) Ovulation occurs 12 to 16 days prior to the onset of menses. After charting cycles for several months, a woman can establish her fertile period by subtracting 18 days from the length of her shortest cycle and 11 days from the longest cycle. The couple must then abstain from intercourse during this time. Another option for some couples is to use a barrier method during the fertile period. An example is as follows:
A woman's menses start every 26 to 30 days.
26 - 18 = 8. 30 - 11 = 19.
The fertile interval is from day 8 to day 19 of the cycle.
(Day #1 is always the first day of bleeding.)
Newer developments which have improved the effectiveness and acceptability of fertility awareness family planning involve using a basal body temperature daily to determine the end of the fertile period, checking cervical mucus for clear copious, slippery status (Spinnbarkeit) to establish the onset and end of fertility, and using a combination of all of the above (symptothermal method).
Failure rates of fertility awareness depend on the motivation and understanding of the participants. They vary from 11% to 47% and are most commonly a result of conscious deviation from the rules of the method. Most accidental (method-failure) pregnancies occur as a result of intercourse prior to ovulation.
Bottom Line: Again, the hard and fast research that has done a strict comparison on spontaneous abortions, pregnancy rates, and the method of birth control have not been fully correlated, or studied in depth. However, given the much higher probability of an egg getting fertilized using the rhythm method, and the fact that 50% or more of all fertilized eggs do not lead to a pregnancy, it follows that more fertilized eggs "die" as a result than if an egg was not ever allowed to become fertilized through other barrier or pharmacological methods. (i.e. No sperm are allowed to contact an egg with barrier methods, and no ova are released with pharmacological methods, totally preventing any type of spontaneous abortion.)