Antonio Conte's men went head-to-head with world champions Germany for an anticipated game. However, no goals were scored, which led to a nail-biting penalty shoot-out.
The Germans won 6-5 on penalties and Zaza was one of the four Italian players to miss from the spot, as the Azzurri were dumped out of UEFA Euro 2016.
Speaking with Sport Mediaset at Milan's Malpensa Airport, the 25-year-old Juventus striker admitted he would never get over his penalty miss.
"I have always taken my penalties like that and was convinced I'd score.
"I did send Manuel Neuer the wrong way, but unfortunately, the ball went too high. It will be difficult, but I want to get over it. I want the Italians to know that we gave everything.
"I'm sorry for letting Italians down. I missed the most important penalty of my career, and it'll be with me for the rest of my life."
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