Inter won an epic second-leg 4-3, with Davide Frattesi's winner coming on 99 minutes in extra-time.
Barca players and coach Hansi Flick took aim at ref Szymon Marciniak in the immediate aftermath.
And Gavi posted to social media on Wednesday: “It’s really tough, especially knowing how close we were. They didn’t want us to reach the final because they were scared of the season we are having.
“It doesn’t matter how many years we’ve gone without winning the Champions League, even if I know it’s hard and I wish we could win it every year.
“I’m sure we’ll win it if we stick together. I know it will come. No matter how frustrated we are now, we can still finish this season by winning three tiles, something nobody expected before the start of the season.”