Grealish is in the England 33-man provisional squad for the tournament, but will have to work to prove his form and fitness to manager Gareth Southgate.
The 25-year-old did play in the final four games of the Premier League season for Villa.
Southgate recently stated that he did not want to "overload" Grealish and risk a setback.
"He is 100% fit," said Smith.
"Firstly, we had to make sure he was ready to play for Aston Villa. We wouldn't take a risk with only four games to go, with a player of Jack's standing and quality, unless it was the right thing to do.
"We all believed it was the right thing to do – for him and for Aston Villa. Then once the season finished, if he gets picked, which hopefully he does, then it's the right thing for England as well.
"If we felt there was any kind of future risk then we would have just shut him down for the rest of the season and he wouldn't have had the chance to go to the Euros."