Yes he can and should. Suppose a president was convicted for a murder he committed prior to being president while president? Now what? Or assuming a president can't be prosecuted while president (that may be the case; the only remedy is impeachment and conviction), suppose it all comes out on tape and he confesses?
Do the research. The times anything like that has come up, the Congress has decided that they don't have the right to do it.
The murder thing wouldn't wash anyway; it's criminal.
Do you not remember all the discussion when the case went to the Supreme Court about whether Clinton COULD be sued while in office?
He lost that suit, i.e., was found guilty and he was NOT impeached for that because it didn't happen while he was in office.
The first time the I word was ever used concerning Clinton was when the Monica story first broke; and that was because of subborning perjury.