The Konigsblauen held Gala to a 1-1 draw in Istanbul in the first leg, but failed to book a ticket for the quarterfinals on Tuesday as the Turkish giant recorded a surprise 3-2 away win in the return.
"It's an enormous disappointment. We had enough chances to win the game, and we gave them their goals. We can only blame ourselves for not having given absolutely everything in the first half. It wasn't enough," Howedes was quoted as saying on the official UEFA website.
"When you look at both legs, then of course the disappointment is all the greater. We were the better team in both games, but Galatasaray scored more goals today and deserved to go through. The things which have made us strong in recent weeks were missing a little today. In spite of that, we've played a great Champions League campaign, and can take many positive things from it."
Schalke will be looking to return to winning ways on Saturday when the club takes on Nurnberg.