Last year's beaten finalist Fritz played some amazing tennis in a thrilling three-set round-robin match in Turin, and after winning the opening set with a decisive tiebreak performance, he could have put Alcaraz away in the second.
"I did a lot of things well in the match, especially the first two sets," Fritz said.
"I think I executed on a lot of the harder things that I wanted to do and kind of almost let it get away from me on maybe some of the things that should be a bit easier to do.
"I think some big points I just wasn't as clinical with the short put-away balls as I would have liked to have been. I just wasn't, like, clinical enough in finishing some points on some really big points."
Fritz forced two break points at 2-2 in the second set, and also came close to breaking Alcaraz at 4-4 but went on to lose 6-7(2), 7-5, 6-3.
"I really believe if I hit my forehand just a little bit better on some shots, I'm up a break in the second and I'm putting myself in a position to serve out the match," the American said.
"The ones that come to my mind, there was one game 15-30, I missed a forehand that I should have at least put away or hit a good enough shot to win the point on.
"I'd say the thing that's frustrating is most of the opportunities that come to my mind were all on me actually having the ball that I want to really attack on, just not hitting it well enough."
Alcaraz took control in the third set as Fritz began to fade, not helped by a long-term knee injury and playing on two consecutive days.
"My knee's completely cooked. There's nothing I can really do," Fritz said.
"I've been dealing with it all year. I've had tendonitis all year long. I started feeling it towards the end of the first set, but it didn't really affect me until the third set."
Lorenzo Musetti, who Fritz defeated in his opening Jimmy Connors Group match, takes on Alex de Minaur, beaten by Alcaraz, later on Tuesday. Fritz faces De Minaur in his final group game on Thursday.