The Northern Ireland international and club stalwart went under the knife in London earlier this week.
Dallas sustained the injury when raced into a challenge with Manchester City’s Jack Grealish on halfway shortly before the interval last Saturday and immediately banged his hand on the turf in agony as he hit the ground.
A Leeds statement later confirmed that Dallas suffered a femoral fracture and would be out for a lengthy period of time.
Marsch said: "Stuart has had successful surgery. It was long - five-and-a-half hours - but they believe that the outcome was very positive.
"They say it will be a long recovery, but believe he will have a full recovery. He has had 400 to 500 messages from family and friends and we know that he is loved and this moment proves it more than anything.
"I don't want to speak for Stuart, but know he has been very appreciative of so many people reaching out and extending their concerns and love towards him.
"I think we can safely say it will be six months, but it could be longer.
"Stuart is such a big part of the group and a fantastic person and player. I think he had the ongoing consecutive record for so many matches and is also an 'iron man'. Stuart will be back stronger than ever."
Meanwhile, speaking ahead of Sunday's trip to Arsenal, Marsch said that they were 'cautiously optimistic' regarding Patrick Bamford's return to the fray shortly.
The availability of the striker - who has not featured since rupturing his plantar fascia in the 3-2 victory over Wolves in March - for a critical end of the season with United fighting for their Premier League lives in the final four matches of the campaign - would be represent a welcome fillip.
Cooper limped out of the warm-up ahead of last weekend's game due to a knee issue.
Marsch said: "Patrick had been on the (training) pitch, so we remain cautiously optimistic with that and Tyler (Roberts), Adam (Forshaw) and Cree (Crysencio Summerville) are all moving in the right direction.
"Liam Cooper and Diego trained today, so we should have our full compliment of players - other than those four or five guys."
On the prospect of Bamford featuring before season's end, the United head coach added: "Yes, I think so.
"I watched him on the pitch a bit today and he was moving really well and doing a lot of ball work and was relatively active and dynamic. So I think we still have two-and-a-half weeks and will go step by step. But we are hopeful we can get him into training by next week."
And on Cooper, he said: "I actually think Liam will play. He was fully in training today and felt good and think he will be available."