Can I omit 'so' in the above sentence? 

If it is possible, Does it have a different meaning from before ?

'So' here replaces the clause it refers to. ie what is being said is
".... the ancient Greeks, who posted sentries at their temples to make sure those who entered entered right foot first, so as not to offend the gods.

Here's a simpler illustration.

Tom cooked spaghetti with mushrooms, very carefully. He did so last night.
What this means is
Tom cooked spaghetti with mushrooms, very carefully. He cooked spaghetti with mushrooms, very carefully, last night.
The use of 'so' avoids excessive repetition.

If you simply say those who entered did so right foot first, the listener will say 'did what?''

Best wishes, Clive