While the Japanese CAP was defeating every wave the Americans sent at their fleet, the successive attack waves were wearing down the Japanese air cover. In fact, cycling fighters off the flight decks was hampering the Japanese ability to spot and launch the next strike.
The real problem for the Japanese was that four carriers were not enough to subdue an island airbase covered by a mobile fleet of three carriers. The Japanese really missed the presence of Carrier Division 5, Shokaku and Zuikaku. In May and June 1942, Carrier Division 5 was the strategic margin held by Japan.
But as others have said, it wouldn't have made any difference in the end, from 12/7/41 they were doomed to lose.
The CAP was also drawn down to the deck because of the attacking torpedo bombers and other waves of B-26s and glide bombing dive bombers from Midway island. There was little or no airborne opposition for the dive bombers from the carriers because of this.