The Ivory Coast international was a half-time substitute against the Gunners in New Jersey, replacing Antony for the second period, but he only lasted 13 minutes.
Amad looked in serious discomfort after a challenge with Arsenal defender Jurrien Timber, with the attacker holding his ankle before he was replaced by Facundo Pellistri.
The 21-year-old was then pictured leaving the stadium on crutches, fuelling suggestions that he could now be facing a spell on the sidelines.
However, the forward has now taken to Instagram to provide an update, with Amad revealing that he initially "feared the worst" but has vowed to be "back soon", insisting that the issue is "nothing serious".
The Ivorian is back at Man United this summer after a successful loan spell with Sunderland, managing 14 goals and four assists in 42 appearances for the Black Cats last season.
Amad has now been involved in his club's three friendlies this summer, starting the wins over Leeds United and Lyon in Oslo and Edinburgh respectively.
The Red Devils will be back in action against Wrexham on Tuesday evening, with an academy side set to take to the field, before the first team features against Real Madrid less than 24 hours later.
Erik ten Hag's side will then round off their tour with a clash against Borussia Dortmund on July 30, before heading back to Europe for friendlies against Lens and Athletic Bilbao.
Elsewhere, two of the club's youngsters Rhys Bennett and Dan Gore have penned new deals at Old Trafford.
Bennett, 19, played 14 times for Man United at Under-23 and youth level last term, while he was also named on the bench for the first team against Nottingham Forest last December.
The teenager is not on the club's pre-season tour due to an injury that he suffered earlier this month, but Gore is currently in the United States, with the 18-year-old making a strong impression this summer.
Gore played in Man United's first two pre-season games of the summer against Leeds and Lyon, while he captained the Under-21s during the 2022-23 campaign.
The midfielder, who was an unused substitute against Arsenal last time out, is widely expected to be in the side that faces Wrexham in San Diego.