No, they don't.
Article II, Section 2, Clause 2 of the United States Constitution, known as the Appointments Clause, empowers the president to nominate and, with the advice and consent of the United States Senate, to appoint public officials, including justices of the United States' Supreme Court.
If the party of your choice cannot hold both the office of the President and a sufficient majority of the Senate to secure confirmation, there is no requirement in the US Constitution that says that a Senate majority must comply with the wishes of a sitting president of the opposing party.
Elections, even down ticket elections, have consequences.