Section 4 of the 25th amendment, which outlines procedures for the removal of a president deemed “unable to discharge the powers and duties of his office.”
As you pointed out the Vice President and a majority of the cabinet must vote to remove the sitting president. The Vice President then becomes President once the House of Representatives and the Senate have been notified.
Unless the president objects to the House of Representatives and the Senate. In this case he is reinstated to the Office of the President.
If the Vice President and the cabinet take another vote and agree that the President should be removed from office the matter will be referred to Congress for a vote requiring a two-thirds majority of both houses.
The 25th amendment should not be taken lightly. It is considered to be a way of removing a President from office that is either deeply infirm or insane. The removal of a President or any other elected official from office is a grave matter and is not an action to be taken lightly. Invoking article 4 of the 25th amendment will be precedent setting and not as easy to accomplish as it might seem.
It should also be noted that Donald Trump the President is the same man as Donald Trump the candidate. How have his actions and attitudes changed in a way that would indicate that Donald Trump is no longer as fit to serve as he was on the day he was sworn into office. Given the fact that the Republican party holds the majority in both the House and Senate at this time obtaining the much needed two-thirds majority in each house of Congress may not be easy at all. Such an action would require that members of the majority party place principle before politics.
If Donald Trump is removed from office prior to the expiration from his term it is more likely to be a result of impeachment for treason, bribery or other high crimes and misdemeanors. Impeachment by the House of Representatives would result in a trial conducted by the Senate and presided over by the Chief Justice of the United States.
Given the President’s recent behavior it would seem quite possible that he may do something of consequence, the nature of which will prevent the Republican majority from withholding action.