The RFU is currently without a permanent chairman, chief executive or performance director, and the executive has previously survived two votes of no confidence.
Thomas is expected to face more calls for his resignation after England's controversial displays both on and off the pitch in New Zealand.
"I'm not in the business on turning my back on conflict," he told BBC Sport. "I've never done it in life and I'm not going to start doing it later in life.
"I am not going to resign over what's gone on because I don't believe I have anything to resign over. Other people may judge it differently and they are entitled to do so."