Salah had to be taken off within 31 minutes in Kiev after a challenge from Real Madrid's Sergio Ramos left him in considerable pain. Liverpool went on to lose the match 3-1 and Salah was considered a doubt for the World Cup.
He missed Egypt's first match at the tournament before starting and scoring in their second and third games, though the Pharoahs lost all three matches.
Salah returned to training at Liverpool on Friday before joining the rest of the squad on their tour of the United States. Klopp says that his star man is in fine fettle ahead of the new Premier League season.
"He has fully recovered," the Reds manager told Liverpoolfc.com.
"We were constantly in contact [during the World Cup] and he was fine. Shoulder-wise, he had no problems. We were in really close contact and it was clear that if he had any problems still then he would have waited for the next game.
"He is a very serious person and he knows that it’s his career. Whatever people want from you, if you can’t, you can’t. But he was completely fine and is completely fine now.
"He looks full of joy and is really happy to be back."
Salah enjoyed a record-breaking debut season for Liverpool last year, winning the golden boot with 32 league goals - the most of any player in a 38-game Premier League campaign.
Liverpool play three matches in America before returning to England, where they begin their Premier League season at home against West Ham on 12 August.