Without silverware since 2005, Arsenal let Van Persie join United for £24m.
"I thought Robin was a mistake," former Arsenal skipper Adams told BBC Sport. "I think he could have played for four or five more years.
"I hope they do go on and win everything. Realistically, I think they will fall short again."
The Holland international still had a year left on his contract but insisted he had no intention of signing a new deal.
"I wouldn't have made that decision [to sell Van Persie]," said Adams in an interview with Saturday's edition of Football Focus.
"Because I do believe if you want to win major trophies you need [the] best players."