Cole has been cleared to play in Friday's World Cup qualifier with San Marino after apologising for his Twitter post on Friday when he branded the FA "a bunch of t***s".
"This is a player with 98 caps, who has hardly ever missed a game for England when he's been fit," said Hodgson.
"He's been regarded as one of the best, if not the best, left-back in the world for years. No-one could question his commitment to the shirt.
"I'm quite relieved this matter has been completely resolved. He (Cole) made it very clear he regrets it. He did it in anger. He immediately tried to rectify his mistake."