If the deed to the house is in both your names then you each have equal rights to the house. He has no right to throw you out just like you have no right to throw him out.
Whether or not you date is really a personal question, not a legal one. I think a more important question is why you haven't initiated a dissolution of marriage action. If you want to date other people, you are sleeping in separate bedrooms and aren't sexually intimate, why not just get a divorce so each of you can move on with your respective lives?
Neither can "throw" the other out of the house. In Virginia, however, you still are legally married and adultery still is a crime so your "dating" could be a criminal offense and certainly grounds for divorce. Since the two of you are living like this anyway, it would seem prudent to discuss the terms of an uncontested divorce and the legal dissolution of the marriage.
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