Manchester United co-chairman Joel Glazer says the club "apologise unreservedly for the unrest" caused by the proposed European Super League.
United, Liverpool and four other Premier League sides withdrew from the ESL on Tuesday after a backlash.
"We failed to show enough respect for its deep-rooted traditions," Glazer said in an open letter to fans.
Earlier on Wednesday, Henry took sole responsibility for the "unnecessary negativity", saying: "In this endeavour I've let you down.
"The project put forward was never going to stand without the support of the fans.
"Over these 48 hours you were very clear that it would not stand.