According to my intuition, 見つめたなり is much better, but 見つめるなり is not incorrect, either. Anyway, let me try to explain this as logically as I can.
I think your friend has oversimplified the rule of relative tense. The tense in a subordinate clause should be relative to the main clause. It can be in ta-form when you refer to something that happened in the past relative to the main clause.
～まま describes the continuation of state that was caused by an action in the past. Therefore it's always used with the ta-form; for example 彼は出て行ったまま帰らなかった, 窓を開けたまま寝てしまった, その時計は壊れたままです.
Does this mean ～なり is always used with the ta-form, too? Let's see what 明鏡国語辞典 says about なり (emphasis mine).
So, なり is not always interchangeable with まま. When なり is preceded by the attributive form (①), it means "as soon as ～" or "the moment ～". When なり is preceded by the ta-form (②), it's interchangeable with まま.
I think your sentence in question is using なり in the second sense. That is, he stared at her, and while he was still staring at her (continuing state), he became motionless. But it's not impossible to read this sentence as "as soon as he stared at her, ...", so 見つめるなり is not incorrect, either.