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Whenever the Office of President of the United States becomes vacant, whether by resignation, impeachment, ill health, whatever, the reason is immaterial.

Naturally, the Vice President takes over the presidency. No problem: sadly, it's happened before.

One of the new President's necessary acts is to select a new Vice-President, to fill out the remainder of the term before the next election. The selection must be approved by both the House and the Senate. However, the Senate is evenly split at 50 - 50.

Now there have been tie votes in the Senate, and they are resolved by the President of the Senate casting a deciding vote if he/she so chooses.

But, Constitutionally, the Vice-President is the President of the Senate! And there isn't one yet! The tie vote is to confirm the selection of a Vice-President!

So, in the absence of a Vice-President, how is a tie in a Senate vote decided?

The Twenty-fifth Amendment states:

Whenever there is a vacancy in the office of the Vice President, the
President shall nominate a Vice President who shall take office upon
confirmation by a majority vote of both Houses of Congress.

Thus, in the absence of a majority the candidate would be denied the office.

The general rule is that a tie vote in either house of Congress loses. Note that the Vice President may vote to create a tie, so that if the senate votes for some measure by 50-49 (one senator being absent or not voting for some reason), the Vice President may vote against it, create a tie, and the measure does not pass.