The Constitution did not have very much to do with Watergate because President Nixon was never impeached. It is very likely that he would have been impeached and convicted if he had not resigned, but that never happened. The main connection between the Constitution and Watergate is the Constitution’s stipulation that a president (or other official) can be impeached for “high crimes and misdemeanors.” That phrase is not explained any further so it is not clear what it means. The House of Representatives would have had to determine whether Nixon’s conduct in the Watergate affair constituted “high crimes and misdemeanors” and, therefore, warranted impeaching him. Since he resigned, they did not have to do so.
The 25th Amendment was very much involved with Gerald Ford becoming president. It was involved in two ways. First, the 25th Amendment says that
In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Since Nixon resigned and Ford was vice president, Ford had to be the next president. Thus, the 25th Amendment mandated that Gerald Ford become president when Richard Nixon resigned. Second, the 25th Amendment also had something to do with the fact that Ford was the vice president in 1974. Spiro Agnew had been elected as Nixon’s vice president in 1968 and again in 1972. However, in October of 1973, Agnew had to resign after having been convicted of accepting bribes. The 25th Amendment also says that
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.
When Agnew resigned, there was a vacancy in the office and Nixon appointed Ford. Therefore, the 25th Amendment was in some way responsible for the fact that Ford was vice president when Nixon resigned.
In these ways, the Constitution had a little bit to do with the Watergate scandal and the 25th Amendment was very much involved in Gerald Ford becoming president.