City won 2-0 at Wembley through goals from Rico Lewis and Josko Gvardiol.
"It's devastating," Gibbs-White told ITV. "I feel sorry for the fans. I hold my hands up and apologise - I should've scored at least one of them. I think as a team we didn't start well, what we were hoping to do. In the second half we came out fighting and we gave it everything, and the support never stopped until the last second.
"We thank them a lot for that, they've been consistent throughout the season. I'm just lost for words. We gave everything. It's sad to go out like that, but we keep going and we still have the Premier League to focus on.
"When the first one hit the cross-bar, I had a feeling it wasn't our day. The second one, I round the keeper, it's a tight angle but I think I should score and that then would've given us energy, everyone in the stadium energy to go again. The boys kept fighting, we showed resilience, we can take that away."
"We just needed one"
The midfielder continued: "If we get one, I feel like we'd have got another one. It was just the flow of the game in that moment. We had a few good chances we could've scored from but all I can put it down to is that it just wasn't our day.
"We had a gameplan everyone believed in. I thought we executed it well in the first half, apart from their goal.
"Anthony is a massive threat for us, and he showed that in the second half. We have to put it behind us now. We enjoyed it while it lasted, but now we have to focus on getting into the Champions League. We have five big finals - we'll give it everything and hope for the best."