Short answer: Not in general, no. Long answer follows below.
Most studies into offending get their data by asking child abuse offenders if they experienced sexual abuse as children (CSA), with many reporting that they did. It has been pointed out that their word is not necessarily reliable and they may have other reasons for stating this.
Not many studies look at abuse victims and survivors in general. A recent Dutch study by de Jong and Dennison (2017) looked at the history of offending amongst 943 child abuse victims, comparing them with 1439 of their siblings, over a 30 year period. They also compared these with a control group of 645 randomly chosen people. So this is not a small scale study.
They found that victims of CSA are at a higher risk of being convicted of a crime, in general, than the control group. Looking at child abuse convictions specifically they found that some CSA victims did have higher rates of convictions but so did their (non victim?) siblings.
What does that mean? It suggests that it is the family environment and experiences, or even genetics, that creates offenders, not the experience of CSA. A question here is whether the siblings were also abused but their abuse was not part of the original court case. de Jong and Dennison looked again at the data excluding convictions for incest and found higher rates of violent offending, traffic crimes and drugs offences for victims but again no higher rates of CSA offending.
If you look at it by gender they found a small but not statistically significant (so it could just be by chance) rise in CSA offences in women who were victims of CSA and no rise in offences by men who had experienced CSA. This last point is very interesting as the majority of convicted abusers are men but there is no evidence, in this study, that abuse creates abusers.
What this study also highlights are the higher risks of criminal behaviour for people who have experienced CSA and this is important too. I believe this is proof that we need better support and information for victims and survivors to help them in their recovery. Society in general will benefit, from reduced crime rates and happier people, as well as the individual themselves.
de Jong, R and Dennison,S (2017) Recorded offending among child sexual abuse victims: A 30-year follow-up. Child Abuse & Neglect 72, 75-84.