The French forward returned to training earlier in the day, after being struck down by acute gastroenteritis last week, but Alonso said the decision to skip Thursday's match was made to prioritise the player's full recovery.
"It was his first day back in boots, his first day running a little, and he did well," Alonso told reporters at Philadelphia's Lincoln Financial Field. "But he's not fully recovered yet.
"We talked about it after training and he prefers to rest and recover properly for when we go through. And if we make it to the round of 16, he'll be available, but he was still in the recovery process and wasn't perfect for tomorrow."
Mbappe, who received treatment in hospital in Miami last week, has already missed Real's 1-1 draw with Al-Hilal and their 3-1 victory over Pachuca, a win they achieved despite playing most of the match with 10 men.
The Spaniards lead Group H with four points, ahead of Salzburg on goal difference, with Al-Hilal on two points and Pachuca yet to secure a point.
Alonso, who recently took the reins at Real following Carlo Ancelotti's departure to manage Brazil, said they had two objectives for Thursday's game.
"The first is to win and finish top of the group. The second is to continue improving as a team ahead of the knockout stage," he added.
While Manchester City are looming as potential opponents in the round of 16, Alonso said his side would not even think about the permutations.
"It's a bad idea to do the math tomorrow," he said. "The only decision is to finish first and give it your all. We're going out to win."
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