No, a police officer does not have to actually witness a crime in order to arrest a suspect. What the officer needs (assuming that we are talking about an arrest without a warrant) is probable cause that the person being arrested has committed a crime.
For example, imagine that a police officer sees someone come running out of a store carrying a bag. The person is running fast and looking over his shoulder. In such a case, the officer would be justified in believing that the person running might well have committed a crime. The officer could stop the person. If the officer found store merchandise on the person an arrest could be made even though the officer did not actually see the person take the merchandise from the store.
Oh, just to add a bypoint, in Ohio what we call Minor Misdemeanor's are not an arrestable offense, unless the person refuses to sign the citation, etc, whether it was committed in the officer's presence or not.
Scroll to the Appendix of Atwater here, it lists State's laws on warrantless misdmeanor arrests and some include felonies. Pot luck some that do not mention felony, should be the same chapter just another section.
In Ohio here an officer can arrest without a warrant for a felony when s/he has probable cause to believe it happened. I am talking about an arrest in a "public place".
If the person is in a home, unless exigent circumstances present themselves, a search warrant in addition to an arrest warrant is needed if the police intend to arrest there.
From scanning some states in the past, I may have come across a few that mandate a warrant for a felony not committed in the officer's presence? You can pot luck scan some.
Per the shoplifting example above, at least for Ohio, that is one of the exceptions to a warrantless misdemeanor arrest. Per our law though, the arrest may take place within a reasonable time from the crime, otherwise a citation needs to be issued OR a warrant can be issued then.
Most misdemeanor's here though require the "in presence" requirement, Felonies, no!