"It was a performance that did not deserve to be associated with us and we are all accountable," said Greening. "This is not taking anything away from Moseley, they played well and showed how to be clinical and take your chances.
"We just turned up with the wrong mental attitude and when things went against us we crumbled."
Welsh looked to be on their way to victory thanks to first half tries from Liam Gibson (2) and Ben Stevenson, but the Exiles lost their way after the break and now face a must-win game against the Whites, whose fixture against Neath last weekend fell victim to the weather.
On Monday, the players and coaching team held an honest meeting at which Greening and assistant coach Mike Friday made their feelings known.
"Mike and I stated how disappointed, let down and ashamed we felt to be associated with the attitude the players showed," said Greening.
"We also said that they had let down Kelvin [Bryon - London Welsh director] and the supporters.
"I also told them that the way they react this week will tell Mike and I what type of person they are, and what this team and its rugby success means to them. I know that result hurt the players and it hurt even more when reflecting upon it on Monday."
Greening is confident, though, that Welsh will have put last Saturday's performance and result behind them when they face Swansea on Saturday.
"We have addressed things and gone back to what is fundamental - hard work, good attitude, good work ethic and honesty.
"The lads have really responded well. Training has been of a high quality this week and the players' attitude has been excellent."
Saturday also promises to be a special occasion as London Welsh welcome Swansea to Old Deer Park in this the club's 125th Anniversary year.
Swansea are fitting visitors to Old Deer Park this season as the Exiles' first game against the Whites was at St Helen's on December 24, 1885 - the year London Welsh was formed.
"We can't wait but first and foremost we want to put these wrongs right and having Swansea is an added bonus," added Greening.