The scientific explanation was not "obviously the Doctor grasping at straws." Star Trek is a secular universe where everything has a scientific explanation; moreover, it's a humanist universe, meaning that there's nothing ultimately beyond the comprehension of the human/sentient mind, just stuff we haven't figured out yet. It would've been going too far if Janeway had actually been converted to a supernatural belief.
I don't think she should have either. Primarily because, well really, it's a pretty horrible method of conversion to have the life of a valued team member hanging over the balance when you're evangelising.
My interpretation of the episode, however, was that it was a statement about faith in general rather than supernatural faith. There's plenty of inexplicable things in science and attributing them to religion is lazy thinking, I say that as a religious person. However, I do think the episode went out of its way to point out that something terribly WEIRD was going on and not easily explained.
For me, I thought the episode was having the Doctor's explanation come off as too "patt" when there was a lot more going on than just a few energy fields interacting. I'm not saying its religious, but I think it was an interesting episode for suggesting thinking outside the box. I guess, for me, Janeway should have stated "something weird happened there but I needed more info to really dissect it."
Then again, I'm the kind of guy who thinks even if Apollo was a god, phasers would work against him.