Can a person sign a contract committing to never taking an ex-spouse
to Family Court?
Basically no. There are important circumstances where waiving a right to sue is not allowed as a matter of law, which often apply. This cannot be done:
(1) if the existence or validity of the marriage is disputed,
(2) in cases where there are (or could be) minor children,
(3) in cases where there are adult children in connection with whom there is child support owed, that is not approved by the court with the child or a guardian for a child approving it,
(4) in the case of alimony provisions that are "unconscionable" as that standard is defined for family law purposes, and
(5) in cases where there was a defect (such as lack of adequate financial disclosure) in connection with entering into the agreement in question.
But, otherwise, it is possible to agree to property divisions in a divorce, attorney fee divisions in a completed divorce, and to alimony awards, that cannot be challenged in court except on the grounds shown above.
Often a family court will retain jurisdiction to resolve disputes that arise to interpret property division and alimony awards that were properly entered into when the alimony award was not unconscionable at the time it comes into being, and to enforce breaches of the agreement. But this doesn't have to be done.
When jurisdiction is not retained, questions of interpretation and enforcement can be brought in a new court.
I've seen divorces where ambiguities in property division language in agreements are interpreted by courts decades later to figure out who owns what property. In one such case there were two courts that might have had jurisdiction to interpret the agreement of the parties, neither of which thought that it had jurisdiction over the case, so to parallel consolidated appeals had to be brought to decide which court had the authority to resolve an ambiguity in the original agreement before either court could get to the merits of the dispute between the parties.
Of course, at some point, all the kids will be grown, there will be no outstanding child support amounts owed, all property will be divided, and in some cases, all alimony awards (if any) will be fully performed. Once that point in time is reached and the statute of limitations for defects in entering into the agreement has run, then no further family court litigation is allowed.
For example, as part of a divorce agreement, my (soon to be) ex-spouse
wants me to put money aside in escrow that my ex can use in the event
that I open up a case in Family Court. The ex is paranoid that I will
continue litigation, as is agreeing to certain concessions if I agree
to this stipulation. When my youngest child turns 21, the escrowed
money will go to me. But if I open a case before then, he wants to be
able to use the escrowed money for his lawyers.
Is this escrow arrangement legally enforceable?
This is probably legally enforceable if it is mutually agreed to, although I've never seen a case in New York specifically addressing that issue. This is a very different question than the title question.