It seems that most of the other folks that would have problems with this type of thing are concerned on grounds of whether the player would somehow mock/stereotype the proposed back-up character.  I must admit - that's not the case, here.  I think the player could probably play a pretty balanced and good female PC. 
And with that admission, it looks like I lost my only somewhat "justified" defense for my actions.
The very fact I'm posting this question here is probably because, somewhere deep down, I feel guilty about asking the guy to change the back-up character's gender.
Please note his request didn't "spook" me.  The Exorcist spooks me.  Clowns spook me.  This just, um, doesn't sit well.  Again, I don't know why.  As one of you kindly suggested, I should talk to my therapist.  But he - er, she - is on vacation this week.  So no help to be had down that path.
Another interesting question was whether I'd feel the same about a female player that wanted to play a male character.  I don't know, to be honest.  I've never run into that situation, and we don't have any female gamers in my current campaign.
I'd say a quick sampling of the replies thus far confirms my fears: I'm weird.  I wish I were more surprised.
If you folks think this hang-up of mine is odd, you'd get a hoot from the fact that I mandate all my players wear muscle shirts during game nights.  The only character classes I allow are artificer and...well, just artificer.  And any time someone rolls a "9" I leap atop the table and hurl feces at them.  
But I suppose those peccadillos belong in another thread.
Whee,
D