The Process for Replacing a Presidential Nominee

Following the release of a videotape in which Donald Trump made degrading remarks about women, many Republicans have withdrawn their support of Mr. Trump and called for his vice presidential running mate, Gov. Mike Pence, to replace Trump as the party’s presidential nominee.

No procedure to force candidate off ticket

There is no established process for the Republican Party to force Mr. Trump off the ticket. Under the Party’s rules, a vacancy can occur only “by reason of death, declination, or otherwise.”
Donald Trump, in the end, has the power to decide whether he remains the Republican nominee. He has been publicly condemned over the last 24 hours by party’s chairman, the Speaker of the House, and the Senate Majority leader, and hundreds of other elected officials. But the ball remains in his court. — Fortune

Trump could resign

Should Mr. Trump decide to resign from the ticket, the Republican National Committee would call a meeting of its 168 members to vote for a replacement candidate. There would be at least five days notice required and members of the committee representing each state’s delegation would cast the same number of votes as they did at the convention. In theory, anyone could throw their hat in the ring, but the Party would most likely rally behind Gov. Pence.

Thousands of votes already cast

However, the list of complications to replace a presidential candidate less than a month before the election would be mind boggling. Ballots have already been printed, and thousands of voters have already cast their ballot in early voting.
In theory, Congress could change the date of the election under Article II of the Constitution which allows Congress to set the dates for elections, but this has never been done. The Electoral College meets on December 19, 2016 to cast their votes. There is no specific requirement that requires that the Electoral College voters have a set period of time between the election and their meeting.
However, there would likely be numerous legal challenges to any attempt to change the election day, and the country would be plunged deep into uncharted waters.

No presidential candidate has ever dropped out after being nominated

In American history, no presidential candidate has ever dropped out of the race after receiving the nomination of his or her Party. However, one vice-presidential candidate died after being nominated (Horace Greeley – 1872), and another resigned from the ticket (Tom Eagleton – 1972).
Both parties have a process in place to account for the possibility of their nominee having to drop out. According to here is what would happen:
Both the Republican and the Democratic parties have rules in their bylaws governing how to fill the vacancy. The Party Chair calls a meeting of the National Committee, and the Committee members at the meeting vote to fill the vacancy on the ticket. A candidate must receive a majority of the votes to win the party’s nod. The same process would happen if the vacancy were to occur after the general election but before the Electoral College meets to vote. If a vacancy should occur on the winning ticket, it would then be the party’s responsibility to fill it and provide a candidate for whom their electors could vote.

Vice Presidential nominee has no special standing before Electoral College vote

If the winning candidate were to die after Election Day (November 8, 2016) but before the Electoral College meets some six weeks later (December 19, 2016), it would be up to the Party to name a replacement.
In other words, the vice presidential nominee on the ticket has no special standing should the presidential candidate be determined to be unable to run before the date the Electoral College meets (December 19) and would not automatically become the Party’s nominee. The Party would convene a special meeting and nominate a new presidential candidate.

Would be a major crisis for the nation

If a candidate should die or otherwise become unable to serve after December 19 when the Electors meet to record their votes, but before Inauguration Day (January 20), the 20th Amendment to the Constitution takes control and states that the vice president-elect assumes the presidency. However, there are murky waters here as well, as there is some debate as to when a candidate becomes the “president-elect.” Most agree that it would be the date of the Electoral College certification (December 19), but some believe it would not be until January 6 when a joint session of Congress convenes to officially accept the vote of the Electoral College and declare the winner. This scenario would no doubt end up in the Supreme Court.
There is no question that the incapacitation of a presidential candidate between the time they are nominated by their Party and the certification of the Electoral vote by Congress would create a major crisis. It is surprising that the procedures are not more clearly laid out by law, and it may be time for Congress to establish a more formal process to account for this possibility before it becomes a reality.