The six-time Paralympic champion suffered the problem with around 10 kilometres to go and was caught by winner Johnboy Smith.
Weir, 43, who only has one Commonwealth gold – the 1500m in 2014 – had a lead of almost three minutes before the issue and eventually finished seventh in the T53/54 event in Birmingham on Saturday.
“I’m feeling despair and I’ve never felt despair in a race before,” he said.
“By coincidence, I was talking with my wife the other day and I said that if I had a big lead I could even afford to change a tyre.
“In the end I decided not to carry a spare, I should have gone with my gut instinct. (Before I punctured) I was flying.
“Where I’ve come from, on a council estate, I don’t give up.”
Smith won in one hour 41 minutes and 15 seconds ahead of Scotland’s Sean Frame and England team-mate Simon Lawson.
But the 32-year-old – who won T54 marathon silver at the Gold Coast in 2018 – paid tribute to Weir.
He said: “I didn’t know he had a problem. I saw him from about 50 yards. I shouted out, ‘what happened?’. I cursed out loud a bit then so if it’s on camera I apologise to everyone. I didn’t want to win by default, but I’ve won gold and I deserve gold.
“If he hadn’t have got that flat I don’t know what would have happened. But I could have closed the gap. It would have been a sprint finish. I won by good circumstances.
“Do not write off David Weir. He is not an old man, he is a supreme athlete.
If Her Majesty the Queen is watching or listening, give him a knighthood. He deserves to be Sir David Weir
“If Her Majesty the Queen is watching or listening, give him a knighthood. He deserves to be Sir David Weir.”
Australia’s Madison de Rozario won the women’s T53/54 race ahead of England’s Eden Rainbow-Cooper and Shelly Oxley-Woods.
Uganda’s Victor Kiplangat won the men’s race in two hours 10 minutes and 55 seconds, despite appearing to go the wrong way at one point.
He still won by one minute 34 seconds ahead of Alphonce Simbu of Tanzania with England’s Jonathan Mellor sixth.
“The people riding the motorcycles were confusing me. They told me to turn back but I still made it to the finish,” added Kiplangat.
Australia’s Jessica Stenson, who has previously won two bronze medals, took the women’s title in two hours 27 minutes and 31 seconds.