The phrase "almost universal acceptance" is correct and usable in written English.
It can be used to describe a situation where something is widely accepted, but not entirely so, indicating that there are some exceptions.
Example: "The new policy has received almost universal acceptance among the staff, with only a few dissenting voices."
Alternatives: "widely accepted" or "generally accepted".
Russell's rejection of idealistic monism, and his arguments in favor of a pluralistic universe, have gained almost universal acceptance, with a few exceptions.
The almost universal acceptance of HIV testing by patients and the stated willingness to accept testing by patients not offered HIV testing, suggested that implementing routine PITC for all patients with TB disease could dramatically increase testing uptake.
His point immediately won almost universal acceptance.
There's almost universal acceptance that Detroit must change.
More fortunate than many pioneers, Lister saw the almost universal acceptance of his principle during his working life.
Given the almost universal acceptance at the time of the necessity of deregulation, that's also an intriguing area in which we need to insert ourselves.
Now there is almost a universal acceptance that we are simply storing too much.
But it isn't her feats of superheroism that give the story its chilly sense of reality; it's her casual acceptance of the almost universal corruption of everyone who lives in her world.
At the core of the program is the almost universal desire of younger children to gain acceptance and approval from their older (and presumably cooler) peers.
However, when analysing the pairwise correlation between variable 1 and 3, tested for different grades of comprehension of natural selection, we see that acceptance of evolution is almost universal (99%).
People in the lowest SES post code area of Playford were slightly more inclined to challenge the validity of the claims in the newspaper report than in the other areas, while in Burnside, the highest SES post code area, acceptance of the claims was almost universal.
