Hi Brett,
I don't know if you would have killed the alternator, since its job is to charge the battery. But you may have fried out a diode somewhere, or possibly the voltage regulator. Could have also shorted out one of the wiring assemblies, which might explain its running for just a few minutes when you kick it over manually
If you don't have the repair manual, you might want to let the dealer do a diagnostic test, since that will pinpoint the problem right away. At least you can get it fixed quicker.
If you would like to get the manual for your rancher, you should find it here...
http://www.atv-411.com/honda-atv-repair-manual-19.html
These manuals have a great trouble shooting section included in them.
Hope this helps...
Ride Safe. Ride Legal. Ride Again... Curtis