What is the ruling of Islamic law regarding a husband who does not spend the night with his second wife during his annual leave, claiming that spending the night is the right of his first wife, and is the second wife entitled to refuse intercourse in this situation?

There is no statement from scholars regarding a wife's refusal to answer her husband for intercourse on the day of another wife's turn. However, the more probable opinion is that she should not refuse, out of consideration for the general proofs that obligate a woman's obedience to her husband in matters of enjoyment, and because intercourse in this situation is not an act of disobedience in itself, but rather due to its transgression upon the other wife's time.

Nevertheless, you must advise him to be just among his wives and clarify that having intercourse with a wife on another's turn is not permissible.