The Rennes midfielder limped out of the friendly against Serbia after just four minutes on Thursday night and was later seen in tears on the bench.
And although he initially feared the injury would keep him out of the tournament in Poland and Ukraine, he has been reassured he should be fit for Les Bleus' opener against England on June 11.
He told L'Equipe: "I cracked up. A lot of things went through my mind.
"When I was 13, I experienced the same thing and I was in plaster for two months.
"I thought of that straight away, that the Euro was over for me."
But he added: "We will let things run their course, and hopefully I'll be ready for June 11, God willing. I'm able to put my foot down on the floor, but I still have to walk with crutches, just in case.
"There is time to heal before the Euros."
France coach Laurent Blanc was also optimistic M'Vila will be okay.
"He has a big sprain on his ankle, but it's not too worrying," Blanc said. "We're hopeful he'll be ready for the first game."
Goals from Florent Malouda and Franck Ribery enabled France to extend their unbeaten run to 20 games and Blanc was satisfied with the improvement he saw after Sunday's patchy 3-2 defeat of Iceland.
The former Bordeaux boss said: "The first 60 minutes were good but after that we stalled. In the third match (against Estonia on Tuesday night) the intention is that we are even better physically."
Blanc was particularly pleased with attacking trio Karim Benzema, Ribery and Samir Nasri.
"With Ribery, Benzema and Nasri, we know we have the technical quality, the individual qualities, but they still had to prove they could play together," he added.
"Tonight [Thursday] they showed they had the right mindset. I hope it will continue."