No. Saying "I'll see you in court" does not create a legally binding obligation to sue. It is a unilateral statement or notice, not a mutual promise agreed to by both parties, and it is also not supported by consideration. It does have other potential legal ramifications, however.
Many statutes of limitations run from the date that someone is aware of the existence of their right to sue someone, and saying "I'll see you in court" is a statement that demonstrates that a statute of limitations was running no later than the time when the statement was made. (The date that a statute of limitations starts to run from is also called the date upon which the claim "accrues".)
Also, in order for there to be a "case or controversy" over which a federal court (and in most cases, a state court) to have jurisdiction to enter a declaratory judgment in a dispute, there must be a bona fide dispute of the subject matter of the declaratory judgment lawsuit. If one of the parties to a declaratory judgment action said "I'll see you in court", this would generally establish the existence of a real "case or controversy" between the parties which would give a court jurisdiction to resolve a dispute between them wiith a declaratory judgment.
Finally, the statement "I'll see you in court" in the context of a discussion regarding an alleged obligation to perform a contract will often constitute a "repudiation" or "anticipatory breach" of the contract which will allow someone alleging that the contract is enforceable to bring a lawsuit for breach of contract even though the act which would otherwise constitute a breach of contract hasn't happened yet.
For example, on April 2, Alice tells Bob that he owes her $100,000 at the end of the month for consulting services she provided to him in March, and he says that her consulting was substandard so he shouldn't have to to pay her anything. In response, Alice insists that he needs to pay the $100,000 by April 30, and Bob responds "I'll see you in court." Bob has probably committed an "anticipatory breach" of his contractual duty to pay Alice the $100,000 sum on April 30, and Alice can bring her lawsuit for breach of contract against Bob on April 3, even though his payment isn't overdue under the terms of the contract yet.