With Inter Milan defender Alessandro Bastoni sent off before halftime, Italy and Bosnia played out a 1-1 draw through to extra-time before the hosts won the shootout 4-1 on Tuesday night.
Afterwards, Gattuso said: "I've finished speaking to the team now, and I can only thank this group of lads. I haven't seen a national team like this in years. It hurts, but we have to accept it.
"I don't want to talk about the referees; we could have scored a second goal, but we suffered. I'm proud of these lads, but it hurts to accept an outcome like this. It takes time, it's a real blow."
He also said: "When you come into a cauldron like this, with the fans pushing, we made a mistake and were left with ten men. But the team held firm, we could have scored another goal. We asked for loyalty to the shirt, today it was the incidents that condemned us."
Asked about his future and the state of Italian football, Gattusoo added: "This isn't the time to talk about my future or anyone else's.
"I don't get into these controversies; I'm a coach. We've been struggling for years, but I'm not the person to tell us what to do; there are better people than me."