Being a woman, I think I can probably provide a more accurate assessment of the situation.
It is NOT true that women will only consider the strongest man at the gym. In fact, most women are turned of by "gym bunnies" because they perceive them as shallow and less trustworthy than average looking men.
It IS ok to say hello to a woman at a gym. She will not necessarily automatically assume you are trying to pick her up, and if she does, is that really a bad thing in light of the fact that you are?
I recall one time I was at the gym and walking down a hallway, one of the other regulars was walking parallel to me a few feet away. Opposite to us was a gentleman about her age who had also be somewhat of a regular, but instead of walking passed us, he stopped right in front of her and said "Excuse me ma'am, I've seen you around here and I'd like to introduce myself."
It was pretty apparent that she was just as smitten with him as he was of her and I imagine it had been love at first sight for both of them.
Next time you see this woman, why not say the same thing?