No, race is categorising people based on how you perceive they actually are, ethnicity is about how they see themselves. I don't know whether 'in relation to genetics' is the be-all-and-end-all, but it's certainly up there somewhere - hence someone can be ethnically 'white' and nationally 'German', or ethnically 'German' and nationally 'American', neither of which actually reflects anything at all on their 'race'. One can even imagine an ethnic German, with American nationality, holding British citizenship.
EDIT: To be clear, the key point with ethnicity is the perception of ancestry, which is inseparable from if not quite the same as genetics per se.
EDIT: To be clear, the key point with ethnicity is the perception of ancestry, which is inseparable from if not quite the same as genetics per se.