Horvath joins Forest as a free agent from Belgian side Club Brugge, where he had made 64 appearances across all competitions since 2016. He also played at Norweigan side Molde, where he was on the books from 2013-16.
The 26-year-old's deal with Forest will be for three years.
At Forest, he'll have competition with keepers Brice Samba and Jordan Smith on Chris Hughton's squad.
"It feels great to be here. Nottingham Forest is such a historic club," Horvath said. "I really like the stadium and training ground and the people have been fantastic. It's a new challenge for me. I'm hoping all my experience can help Forest.
On the international level, he has seven appearances for the USMNT. In June, he came on for the injured Zack Steffen in the CONCACAF Nations League final against Mexico, making a key penalty kick save late in that match to preserve the 3-2 win.
"We are delighted to be signing somebody of Ethan's calibre. He has plenty of valuable experience in Europe with clubs of good pedigree and in European competition," Houghton said. "He will give us valuable competition in the goalkeeping department and we look forward to working with Ethan."
Forest finished in 17th place last season.