He returned to training with Martin O'Neill's squad earlier this week and was able to play a full part on Thursday as their attentions turned to Sweden on June 13.
"I should be fine," said the 25-year-old. "I've trained with the team the last couple of days and hopefully now I can step it up.
"It's obviously good to be back training again with the squad and hopefully now I can kick on and step up the fitness.
"It's hamstring, groin - it's a bit of both, to be honest - but as I say, I am back training and delighted to be back with the squad."
Ireland go into the first game in Group E on the back of three games without a win – including the recent friendlies with Netherlands and Belarus – but McCarthy insists morale remains high within the camp.
"There's a massive excitement about the place, but we'll take it a game at a time and we won't get too carried away," he added.
"Obviously the boss and the likes of Roy [Keane] and the backroom staff, they know how important it is to not look too far forward.
"We have got a tough group, don't get me wrong, but we'll take it a game at a time. Each and everyone one of us who is going on the plane and travelling to France, we know it's going to be tough, but there's a real excitement about the place and we are looking forward to it."