Does the US President have the power to appoint any official to his cabinet a Supreme Court justice or an ambassador without approval from Congress?

No. Article II, Section 2, Clauses 2 and 3 of the US Constitution authorizes the President to nominate members of the cabinet, Supreme Court and federal courts, ambassadors and certain other government positions with the "advice and consent" of the Senate. The Senate must confirm the nominee by a simple majority vote in order to complete the appointment.

This is true even of recess appointments, which allows the President to place a nominee directly into office if Congress is out on a long recess, but this only defers the approval process, it does not eliminate it.

Article II, Section 2, Clauses 2 and 3

[The President] shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur; and he 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, and which shall be established by law: but the Congress may by law vest the appointment of such inferior officers, as they think proper, in the President alone, in the courts of law, or in the heads of departments.

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.