I would say that he loved her, not only wanted her. (Don't wait for an explanation, it's just the gut feeling, and the thought that there must be something good in Gríma even in his present condition.)

I would, as well, say that he wouldn't rape her. He would try to have her, of course, but on a more sophisticated and less brutal way.

He knew she didn't love him, he knew she despised him. I think he would maybe have poisoned her mind with his words, trying to make her think that the men of the House of Eorl were her enemies, because they "held her in a cage" and that way he would have maybe gained her appreciation, then her friendship and maybe then her love, or she would have (")realised(") that she was alone, and sought comfort from him, the only one who treated her as an equal and the only one she had anymore (if she had in her mind turned against the house of Eorl).

Another point, which I think, no one has yet mentioned. In somewhere deep buried in his heart, Gríma had some affection, or appreciation for the House of Eorl and King Théoden, and especially for Éowyn. He wouldn't ruin her: I agree the ones here who say that for him Éowyn was something too pure to touch without her agreement. (Her being a prize after Saruman had won is a different thing.) I'd say that he waited for the moment when her heart would turn to his (which maybe would have happened, if the men of The House of Eorl had continued not to understand her and thus she had become to believe the half-truths he told her).

The fear of Éomer, which Noxomanus mentioned, is also a good point in my opinion. Though Gríma maybe guessed, that Éowyn wouldn't tell her brother, but he couldn't be sure.

What comes to the physical fight between Gríma and Éowyn, I would say it would be tough. Gríma was weak for a man (and wasn't he described as old? Age lessens strength.) and Éowyn was strong for a woman. Still she was slender; I think strength wasn't her best ability as a warrior, skill and dexterity more probably. And, as said before, the sword-skills don't matter in a fight like the one we're talking about. If I had to say, which one would win, I would go for Éowyn. (And it's not the little feminist inside me - I swear :D)