Webber has so far been the only driver to publicly criticise the 68-year-old in the wake of lurid revelations into his private life that were published in the News of the World.
The Australian claimed the furore surrounding Mosley had "brought the sport into disrepute", adding that F1 "simply cannot have scandals of this type".
Since the story came to light, there have been calls for Mosley to resign from various bodies.
That has resulted in an extraordinary meeting of the FIA general assembly being called, to be held in Paris on June 3 at which a vote of confidence will be taken.
Prior to that hearing in 12 days' time, Mosley is making his first public appearance at a grand prix this weekend, even though he is not officially representing the FIA.
However, Mosley yesterday held meetings at his private offices with the team principals of Ferrari and Renault in Stefano Domenicali and Flavio Briatore.
He is also due to appear in the Formula One paddock at some stage today, and for Red Bull star Webber, the reaction he receives is likely to be crucial.
When the question of Mosley was put to Webber, he replied: "To be honest, I'm not going to bed thinking about what's going to happen. I'm not losing a huge amount of sleep over it."
"He's big enough and old enough to make his own decisions for himself, and what will be will be this weekend.
"We'll see what happens, and we'll see what happens on June 3 as well. We'll see how we feel with his presence, and see how people handle all that.
"It's obviously the first time it has happened since the scandal occurred, and we'll see what the reaction is to having him here in person."
Webber concedes to disappointment at being a lone voice in the wilderness, and that no other driver has supported his stance.
"I'm disappointed, but not surprised," he remarked.
"Most guys say 'no comment' because it's easier, but if someone asks you if it is good or bad, why can't you say what you think?
"It's only an opinion. It's not earth-shattering."
BMW motorsport director Mario Theissen believes Mosley's return to Formula One will not overshadow what is regarded as the blue riband event of the motorsport calendar.
"I don't think so and I don't expect that," replied Theissen when asked about the thorny issue of Mosley.
"I don't believe it will be the dominant issue this weekend."
It has been suggested a number of F1 personnel are keen to avoid stepping into the spotlight focusing on Mosley this weekend, and do not want to be photographed with him.
Although Theissen will not deliberately shun Mosley, he added: "I am not looking for him."