Forest were beaten 1-0 by the Premier League Hornets on Saturday and Wilson was at fault for the goal that they conceded.
The former Celtic defender failed to clear, and he was punished when in-form forward Odion Ighalo netted the winner.
However, Gardner was quick to praise the centre-back after the match.
He said: "The lads are gutted. We had the opportunities to win it and I thought we were solid at the back. The defence were brilliant for 90 minutes and one mistake has cost us the game. I thought Kelvin was brilliant and that’s how cruel football is.
"But we’ve all got to keep our heads up. We’re absolutely gutted but we can take positives out of this. We’ll be back in training next week. We’ve got to keep our heads up, train well and get ready for Leeds.”
Forest put in an excellent account of themselves against Watford, in what was an even contest at the City Ground.
Either side could have come out on top but it was Watford’s Premier League class which eventually told.
Forest had their chances in the fixture, but Ighalo’s ruthlessness sent Quique Sanchez Flores’ side into the last 16.
The defeat was Forest’s first for 14 matches, and they will be determined to get back to winning ways in their next fixture against Leeds United.