The President can, in fact, recess appoint a Supreme Court justice per Article II, Section 2, Clause 3[0].
Since the George W. Bush administration, Congress has used pro forma sessions[1] to prevent recess appointments. Both the House and Senate would have to agree on a time to adjourn Congress per Article I, Section 5, Clause 4. If they disagree but one of them wants to adjourn, the President can adjourn them under Article II, Section 3. But no president has ever done this. President Trump talked about doing it to ram through his appointments both in 2020 and last year during the transition period. But so far it hasn't been deemed necessary because the Senate has, surprisingly to me, confirmed his cabinet in a timely manner and without significant pushback even on the less conventionally conservative choices like the DNI and the HHS Secretary. In all likelihood, the threat of adjourning Congress and of using his billion dollars plus of fundraising for 2026 to primary uncooperative Republican members of Congress has forced them to largely fall in line for now.
Recess appointments to the Supreme Court were common in the old days when the Court was less politically contentious. Justice William J. Brennan was recess appointed by Eisenhower and later confirmed by the Senate. A recess appointment who is not confirmed by the Senate would be null and void at the start of the next Congress on January 3rd of the next odd numbered year. I doubt any president would recess appoint a Supreme Court justice today both because it would be likely derail their nomination and also because a recess Justice might get to hear at most 1 term of cases depending on timing. Recess appointing somebody to run the FDA or the Justice Department or even to be a district court judge would be much more useful to a President's agenda.
[0]: "The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session."
[1]: These are sessions where they immediately adjourn by unanimous consent after doing the formalities to open the session. C-SPAN broadcasts them live and they only last a few minutes at most.