The NCC Staff at The Constitution Daily writes (abridged):
The Constitution, in Article II, Section 2, says that the President “shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the Supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for.”
The Appointments Clause allows the President to make nominations for appointed positions like cabinet officers, but the Senate controls the process, including the rules that allow a nomination vote to get to the full Senate floor.
If the Senate isn’t officially in session, the President does have the power to appoint officers directly using his recess appointments powers, but modern parliamentary techniques used in the Senate rarely allow an opportunity for such appointments.
There is little doubt that the direct nomination and approval of cabinet officials falls under the Appointments Clause while the Senate is in session. In the 1st Congress, President George Washington nominated Alexander Hamilton, John Jay, and Henry Knox to his first cabinet, and the Senate approved the nominations by a simple majority vote.
Since then, the Senate has been the gatekeeper of cabinet approvals.