If there was a civil custody proceeding, where non-parent was awarded
custody from parents (not a matrimonial custody dispute), and the
grounds for awarding/stripping custody was child abuse by the parents
(meaning that the court found the abuse to be a fact), does that
necessarily mean that the parents would be prosecuted for child abuse
as a consequence?
No. No one in the civil custody proceeding is obligated to report it and often no one would. Also, there are two possible escalations from a child custody proceeding.
One is an action initiative by Social Services to terminate parental rights for abuse or neglect in a civil proceeding (which does not require proof beyond a reasonable doubt and requires an effort to create a plan to restore the family if feasible that fails before going to completion).
The other is a criminal prosecution for abuse and neglect, which requires proof beyond a reasonable doubt and has the consequence of criminal penalties, usually including incarceration.
A custody determination can also be made based upon a high but not necessarily even imminent risk of future harm to a child, even if the abuse or neglect has not actually yet occurred, while the other two escalated options require something specific to have actually harmed the child.
Any testimony or evidence in the child custody case would be admissible in the other two kinds of cases.
But, realize that there are a couple of ways that a custody determination could arise in circumstances of suspected abuse.
One is that it could be fully tried on the merits based upon a presentation of evidence including sworn testimony which would be good evidence in a future hearing.
Another is that the parties could, before the hearing or early on in a hearing, have had an off the record discussion (with or without a mediator) in which the abusive parent basically conceded that the outcome would be bad for her seeing the exhibit notebook and witnesses that the other side had on hand to present evidence, or she could have just decided that she didn't care enough to fight it. Either way, the parties could have gone to the court and simply stipulated to a resolution in the form of a custody order, rather than litigating it on the merits. The mere stipulation for unstated reasons would typically not support a termination of parental rights case or a child abuse or neglect criminal case - evidence of settlement generally cannot be used to provide "liability".
In any case prosecution of either of the other two escalated cases (termination of parental rights or a criminal case) is in the complete discretion of social services, law enforcement and prosecutors. They can decline to prosecute for any reason or no reason at all. Common reasons not to prosecute include lack of resources, opposition of the victim or the victim's non-abusive family members to these courses of action, and a low level of risk of future harm.
The actual question is slightly the opposite: does the fact that the
parents were not criminally prosecuted for child abuse necessarily
mean that the custody case did NOT involve child abuse as the accepted
reason for removing custody from the parents? Or is lack of criminal
prosecution not a sufficient factual base to make that assumption?
No. Lack of criminal prosecution does not imply in any way that there was not child abuse.
A custody determination that de facto prevents future abuse may make the case a lower priority for social services and law enforcement, but it does not prevent such a case from being brought.