The President can never appoint someone to fill a Senate vacancy. The U.S. Constitution allows the Governor of the state to which the vacant seat is entitled to make a temporary appointment to fill the seat until a special election can be held, but only if the state legislature has passed a bill allowing him/her to do so.
Actually there is one Senate vacancy for which the President appoints a replacement: the U.S. Vice President, who constitutionally is also the President of the Senate. Vice presidential appointments require the approval of both houses of Congress.
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The Speaker of the house becomes president
The U.S. vice president is first in line to fill the president's unexpired term if the office becomes vacant for any reason.
No, if the President is removed, retires or dies the Vice President will become President. If the Vice President can't fill the office for some reason then next in line is the Speaker of the House of Representatives.
Amendment 25 gives a process for filling a vacancy in the office of vice president.What happens is that the new President names someone to fill the position and if both houses of Congress approve, he becomes vice president.
The VP does not preside because in the presidential line of succession if the President was removed, the VP would fill the role as president, meaning that the VP has enough motive to see the President removed from office.