There’s a great deal that’s not specifically spelled out in the Constitution, and that’s why we have a Supreme Court.
For example, in Article II, Section 2 of the Constitution, it says, “The President […] shall have power to grant reprieves and pardons for offenses against the United States, except in cases of impeachment.” And it’s true that it doesn’t specifically say that the President cannot pardon himself.
However, just because there’s not such a restriction on this right of the President doesn’t mean that it should be assumed it doesn’t exist. It also doesn’t say that the President isn’t allowed to murder his enemies, either — and that doesn’t mean he can. It’s always been a common standard of law that you can’t be your own judge, and issuing yourself a pardon falls in that category.
This would not be the only right outlined in the Constitution that has restrictions. Most of the Bill of Rights has some kind of restrictions — for example, your right to free speech doesn’t give you the right to deliberately use it to incite a riot or hazardous situation, and your right to bear arms doesn’t mean that you get to walk into your local supermarket carrying a rocket launcher.
If push came to shove on this issue, it would wind up in the Supreme Court, and they would be very careful before they made any decision that would allow this. For a very good reason — if they allowed Trump to pardon himself, they would be allowing all future presidents to pardon themselves (regardless of party), essentially giving them free reign to get away with whatever they wanted until Congress decided to impeach and remove them.
But let’s pretend for a moment, that Trump pardoned himself for any and all crimes or misdemeanors committed prior to the date of the pardon. A big get-out-of-jail-free card. And let’s pretend that the SCOTUS backed him up on this for whatever reason. What do you think would happen? In the first place, the act of pardoning himself would be a de facto admission that he had done something requiring a pardon, an admission of guilt. He could still be impeached, as his self-pardon would have no effect over that. And he could be removed from office. But because of the pardon, he would not be charged with anything in our civil justice system, nor would he be tried in a civil court, nor would he go to jail.
But he’d still be toast — much in the way that Bill Cosby is. Everyone would know he was a guilty man, and the public would shun him accordingly.