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The Reds said goodbye to captain Gerrard with a 6-1 thrashing at Stoke to finish sixth in the Premier League.
"I've always said if the owners want me to go, then I go," said Rodgers, who has won two of his last nine games.
"For everybody connected with Liverpool, we are all embarrassed by that result."
It was the first time in 52 years the Reds had let in six goals in the league but Rodgers, 42, insisted he still has "a lot to offer the club", despite winning no silverware during his three years in charge.
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Liverpool must also now go through two qualifying rounds if they are to compete in the group stage of next season's Europa League.
"A lot has happened this year which has made the job difficult," added Rodgers, who guided the Reds to second place last season.
"Last season, when things were working well, we had the support of everyone but performances like today do not help that and I fully understand that.
"The fans deserve an apology. They were angry at half-time - and rightly so - because that was awful, absolutely awful.
"They have every right to be angry and frustrated and of course I take full responsibility for that as the manager."
Rodgers faces a big rebuilding job in the summer and, as well as Gerrard joining LA Galaxy, there is speculation about the future of young forward Raheem Sterling.
Rodgers dropped Sterling, 20, to the bench at the Britannia Stadium and Liverpool found themselves trailing 5-0 at the interval after a terrible display.
All five goals came in the space of 23 minutes from Mame Biram Diouf, twice, Jonathan Walters, former Liverpool midfielder Charlie Adam and Steven Nzonzi.
Gerrard marked his farewell with a consolation goal but there was still time for ex-Liverpool striker Peter Crouch to head a sixth for Stoke five minutes after coming on as a substitute.
Asked about his rebuilding plans, Rodgers added: "There's an awful lot of work to do and the job is to now go and fix that, and make sure we come back with a motivation greater than ever to push on next season.
"We've finished the season in sixth, which is where we deserve to finish.
"We have to now start, today, on the climb to get back up again."

The wreck of the 3rd Century trading ship Asterix was discovered in the mouth of St Peter Port harbour entrance in the 1980s.
The timbers are to go on display as part of an exhibition about a Celtic hoard of 70,000 coins.
The hoard was found in a field in Jersey by two metal detector enthusiasts in 2012.
The timbers have been undergoing conservation work at the Mary Rose Trust in Portsmouth.
Val Nelson, Jersey Heritage Trust Registrar, said the timbers from the Asterix would be on display with a scale model of the original ship.
She said: "This is the first public airing for the timbers, the restoration work has just been completed.
"The coin hoard will be the major part of the exhibition, but we have borrowed objects from Guernsey, Normandy and Brittany to put the hoard in context and show what was happening in this part of the world at the time.
"As well as the timbers, we will have a very special piece from Normandy that is considered a French national treasure."
The timbers had to be lifted into the museum gallery, on the third floor, using a large crane.
The vessel is the "biggest object from Roman Britain and the most intact sea going ship [of its age] found outside the Mediterranean," said Jason Monaghan, director of Guernsey Museums, which owns the wreck.
The whole ship, for which a permanent home has yet to be found, could not be included in the exhibition as it is too large at about 5m (16ft) wide and 17m (56ft) long.
The exhibition opens at the Jersey Museum in St Helier on 2 June. It will visit Guernsey in 2015.
Liverpool boss Brendan Rodgers accepted his future at the club is in doubt after Steven Gerrard's final match ended in a humiliating defeat.
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Timbers from a Gallo-Roman wreck found off Guernsey will go on display in Jersey later as part of an exhibition.