Let's assume that a president of U.S. who has been elected for the first term isn't nominated by his own party in the second term then what will happen?
Has this happened before in U.S. elections?
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Unless there's some special-situation trick in the question I'm not detecting, what happens is rather UN-surprising, and according to usual rules for Presidential elections:
Incumbent president continues serving the remainder of their term till new term inauguration (unless they are so ashamed they immediately resign)
The candidate who won the party primary faces off the winner of another party's primary (as well as any other eligible candidates with ballot access) in general Presidential election.
The winner of that election - whichever party they are in - is sworn in as next President during inauguration.
However, some things might happen politics wise that are harder to predict but are plausible. E.g.
The fact that incumbent president loses the primary may indicate a fractured party base which is probably going to reflect in poor turnout which in turn will reflect a poor showing in general election for that party, so their opponent from another party is more likely to win the next term.
The fact that the winner isn't the incumbent means they don't have what's called "incumbent advantage" - and the fact that they trounced the sitting President in a primary means there's a high likelihood the President won't go out of their way to help the upstart win using their Presidential incumbent advantage. Once again, their opponent from another party is more likely to win the next term.
The fact that the winner isn't the incumbent MAY (not always) mean they are not a popular incumbent. Which means they probably have low general favorability rating, which, as per 538, correlates to success of general election for the party (more so for midterm Congress elections but that's beside the point). Once again, their opponent from another party is more likely to win the next term based on fundamentals.
This kinda happened, though not precisely.
In 1912, President Theodore Roosevelt lost the primary to Taft. He wasn't an incumbent at the moment (Taft was), but he was a very popular party leader and two-term President at the time; Taft was his personally-groomed successor and friend.
Having failed the nomination, he huffed off (as he strongly disagreed with Taft's policies) and founded an independent third "Bull Moose" Party, which caused a split in the Republican electorate and thus ensured election of Woodrow Wilson.
While not an exact situation the question mentioned, it is worth noting the possibility of the same outcome (the incumbent loser of the primary runs as an independent/third party; splinters their party vote and hands the election to opposing party. All repeat after me: Once again, their opponent from another party is more likely to win the next term.
Franklin Pierce was the 14th president and is best known for his support for the Kansas-Nebraska Act of 1854, which ultimately led to violent political clashes dubbed "Bleeding Kansas". These clashes were an early prelude to the American Civil War, and the fight over this bill ultimately destroyed the Whigs as a national party and also helped result in the creation of the modern-day Republican party.
When he ran for re-election during the the 1856 Democratic primary, Pierce failed to secure two-thirds of the votes at the Democratic National Convention through 15 different ballots, never once receiving a majority or even the most votes. He directed his supporters to vote for one of his primary challengers Stephen Douglas in a bid to keep the nomination from another challenger James Buchanan. Douglas also eventually dropped out for fear of fracturing the party before the general election, resulting in Buchanan receiving the nomination. He was a popular choice at the time as the favorite son of the large state of Pennsylvania, and he also happened to be outside the country as Ambassador to the United Kingdom during the debate over the Kansas-Nebraska Act, so he never had to take a firm stand on it one way or the other.
Buchanan was endorsed by Pierce after the convention, though they did not exactly see eye to eye on policy. Buchanan was able to win the general election over two opponents, John C. Frémont and former Whig President Millard Fillmore.
According to NPR, there have been a few occasions when an incumbent President was denied his party's nomination. However, the last time this happened was in 1884, and in four of the five cases they cite, the incumbent President was only in his position due to the death of his predecessor. Only President Pierce was elected President (1852) and failed to win re-nomination four years later. President Pierce continued to serve the remainder of his Presidential term, and supported his party's nominee (and eventual successor) James Buchanan during the general election campaign.
It would be very difficult for an incumbent President who fails to get his party's nomination to stand as an independent. This is because many states have what are known as "sore loser" laws, which prohibit someone who has lost their primary election to stand in the general election as independents. However, in some states they may be nominated by a smaller party, as Gary Johnson was the Libertarian candidate in several elections.