Basically, yes, there's a correlation. Keeping in mind Oddthinking's comment on the value of such non-random samples, this is where most of the correlational data comes from, e.g.:
A history of childhood cruelty to animals and contemporary patterns of
chronic interpersonal aggression has been documented in assaultive incarcerated
offenders, perpetrators of sexual homicides, rapists, and child
molesters (Felthous, 1980; Felthous & Yudowitz, 1997; Kellert & Felthous,
1985; Merz-Perez, Heide, & Silverman, 2001; Ressler, Burgess, & Douglas,
1988; Tingle, Barnard, Robbins, Newman, & Hutchinson, 1986), making a
case for the potential prognostic value of childhood animal cruelty.
As to what is the ultimate cause however, it's nowhere near as simple as MSalters' answer makes it to be, i.e. [lack of] empathy is not the only cause, although it does play a role:
animal abuse was negatively associated with affective empathy and national culture
Childhood animal cruelty (CAC) is also correlated with [prior] exposure to interpersonal and animal violence (same source):
Rural adolescents were more likely to abuse animals and had higher exposure to domestic violence, which (in turn) was associated with more animal abuse.
And as to whether CAC has better predictive value than these other associates... one review says no. Quoting from the first paper:
In a recent review of research on children who are cruel to animals,
McPhedran (2009) concluded that the home environment is generally a better
predictor of the development of adult violence than childhood cruelty to
animals and we should not infer that animal cruelty leads to other antisocial
behaviors throughout development. McPhredran suggested that the context
within which cruelty to animals occurs is most likely the common factor, and
that the complex nature of abusive home environments and experiences that
contribute to behavioral problems and interpersonal violence should be an
important target of our interventions.
McPhedran, S. (2009). Animal abuse, family violence, and child wellbeing: A review. Journal of Family Violence, 24, 41–52.
Alas all review works in this area seem to be of the narrative kind, so the quality of inference (one way or the other) is likely to be biased.
McPhedran's review seems to be talking at some length about what appears to be the New South Wales study that prompted your question, and Oddthinking's comments (on the lack of controls etc.) are reflected in McPhedran's review as:
Although Gullone and Clarke (2006)
contend that there appears to be a greater likelihood that
people alleged to have abused animals will engage in
offenses against the person when compared to all alleged
offenders, they did not specify the order of offending. Thus
it cannot be concluded that animal cruelty preceded other
It must also be noted that although 25% of animal abuse
offenders committed offenses against the person, this
entails that 75% of animal abusers did not commit offenses
of that nature. It is acknowledged that offenses may have
been committed but not detected by police, but it should
also be considered that this appears an insufficient
explanation to account for the entire 75% of animal abusers
who did not commit offenses against the person. Therefore,
the findings simply support existing evidence that animal
cruelty, and particularly the deliberate infliction of unnecessary
physical pain or suffering, may occur in conjunction
with a wider range of antisocial behaviors.
Similar patterns of generalized criminality among animal
cruelty offenders have been noted in New South Wales,
Australia (Gullone and Clarke 2006). The range of criminal
behaviors performed by persons in New South Wales Police
databases with a record of animal abuse averaged four
different types of criminal offence. Assault, followed by
stealing, driving offences, and domestic violence were the
most common forms of crime committed by animal cruelty
offenders in New South Wales who had other criminal
charges. However, the New South Wales animal cruelty
offender statistics were not compared against a wider
sample of offenders without a history of animal cruelty.
Therefore, it cannot be concluded that patterns of generalized
criminal offending among animal cruelty offenders
differed from patterns of generalized criminal offending
either overall or among offenders who did not commit
McPhedran also critiques the lack of operational definition for animal cruelty in some of the research in this area.
And this is in fairly strong contrast with a paper by Simmons et al. (2014), which although is not centered on [interpersonal] violence, does make pretty bold claims as to the predictive value of animal cruelty and appears to have decently large and representative samples:
The objective of this study was to document the long-term relationship between youthful animal abuse and a variety of problem behavior outcomes later in life. Data were used from a national, longitudinal, and multigenerational sample collected by the National Youth Survey Family Study, which assessed families across 27 years from 1977 to 2004. The analytic sample consisted of 2538 individuals who were analyzed using multivariate ordinary least squares and logistic regression modeling that controlled for important demographic factors. Hypotheses were tested across two generations separately showing that a history of animal abuse does, indeed, predict later problem behaviors, including serious offending, marijuana use, other drug use, alcohol use, and deviant beliefs. Depending on the outcome examined, each model accounts for 5–34% of the variation in respondents’ problem behaviors. Within each model, animal abuse was often one of the strongest predictors.
So what to conclude from all this? Low quality primary studies leave substantial leeway for investigators or reviewers to interpret the data according to their own predispositions or agendas. Also, since all these studies are observational, you can never rule out all confounders etc. What's the best predictor may depend on what you included or not in the model etc. So big grain of salt prescribed for the reader and perhaps more research needed.