Liverpool now have just two established forwards in Luis Suarez and the newly signed Fabio Borini, after Carroll was allowed to join West Ham United on a season-long loan.
Rodgers has seen his team pick up just one point from their first three matches of the English Premier League season, with the latest reverse a 2-0 loss at home to Arsenal on Sunday.
Asked by reporters if Carroll would have left had it been known a replacement would not be secured before the transfer window shut, Rodgers replied: "No."
Clint Dempsey had been heavily linked with a move to Anfield, but Liverpool failed to agree a deal with Fulham and he instead joined Tottenham Hotspur.
"I was very confident I had a deal sewn up but it has gone and I can't do anything about it," Rodgers said.
"There's no point me crying about it or wishing we had or hadn't done this or that."
Rodgers was coy when asked about the possibility of signing former player and free agent Michael Owen, saying only that he would look at any opportunity to strengthen the squad.
"Hopefully, in January and the next window, we will make the transitions to get a group that can compete," he said.
"That was always my intention at the beginning of the season. I wanted to build a squad that could be competitive."