You wouldn't say "He repaired his car the whole day" because as one of the commenters said, simple past is used to express that an action was completed.
If you say "He repaired his car yesterday" that means he completed the action and his car is now fixed.
If you want to express that he was engaged in the activity of repairing his car for the entire day (and may or may not have successfully completed the repair), the most natural way to say that would be:
He spent the whole day repairing his car.
"He was repairing his car the whole day" is perfectly fine, and I think would be a natural answer to a question like "Why wasn't John at the softball game yesterday?" In response you might say, "He couldn't make it. He was repairing his car the whole day."