@oristarA,
Quote:I wonder whether the omitting is grammatically fine.
Has the meaning been kept the same?
Of course. Would I have suggested it otherwise?
Here's the amended sentence:
I believe that Clinton didn't make the promise limited to religious context.
Here's my suggestion as a slight improvement:
I believe that Clinton didn't make the promise on religious grounds alone.
or,
I believe that Clinton didn't make the promise based purely on religious grounds.
Is that what you want?