Try it and let us know.

A few thoughts...
If you look at a spinning rod, you'll notice that the first souple guides from the reel are quite large, and get smaller as you work your way to the tip. There is a reason for this. As the line comes off the spool, it spins. Having the larger eyes prevents the eyes from "choking" your cast. The smaller eyelets on a fly rod will choke your cast, drastically reducing your distance.

Next is balance. Fly reels aren't reeled in on every cast. You strip line, then cast again without reeling up your line. You don't fight fish with the reel either (although a good fly reel with a drag helps). Most of your "drag" while fly fishing comes from your fingers on the line. I would think that reeling a spinning reel that is attached at the very butt of the rod would be very awkward to reel.

There are spinning rods on the market that have similar length and actions to a fly rod, which would be better suited for what you're talking about and would eliminate the problems above.

Now, I have seen people use fly rods, with fly reels spooled with mono, before. Casting is different from fly casting and from spin casting, and is best for short ranges. Also, there are "center pin" reels that are great for working a drift while fishing rivers and streams. The center pin reel is free spooling and allows line to pay out during a drift, giving your bait a very natural presentation. These outfits are a favorite among steelhead fishermen in our area. Just another option to consider.

There are always options. You can give the spinning reel on the fly rod a try, but I would think that there would be other, much less awkward, options to try first.