A man assaulted a woman while in Mexico. She did not press charges there and they returned to the U.S. Can she press charges against him in this country for something he did in Mexico?
Criminal cases can generally be brought only in the jurisdictions where they were committed or where they caused injury. Only the jurisdiction that has enacted the relevant criminal laws can enforce its own laws, which presumptively apply only in the territorial jurisdiction of the place enacting them.
In the case of a criminal assault case, where the assault occurred and caused injury solely in Mexico, criminal charges could not be brought in the United States.
The Mexican government could seek to have the perpetrator extradited to Mexico from the U.S. through a process set forth in a treaty between the U.S. and Mexico.
This said, while criminal charges could not be brought in Mexico, the injured woman could sue the perpetrator of the assault for this act in a tort lawsuit for money damages including but not limited to compensatory economic damages, non-economic damages for pain and suffering, and exemplary or punitive damages for intentional misconduct, in the United States brought in the county where the defendant resided in the U.S.
This is because U.S. courts of general jurisdiction have "general jurisdiction" over civil claims committed someone who is domiciled in its territorial jurisdiction over any legal claim wherever in the world it arose.
I've personally brought a case like that for an incident arising in France. It wasn't dismissed for lack of jurisdiction, but it is also very hard to prove a case like that in a U.S. courtroom because securing the presence of witnesses other than the perpetrator and victim, and securing other evidence, is very challenging.
Often a U.S. court will not have subpoena power over foreign witnesses except through depositions taken with "letters rogatory" approved by the local courts where the witness is located, which is an expensive and slow process and it rests to some extent in the discretion of the local courts.