Both whtietail and red deer are ungulates. The term simply means two toes or split hooves.
Elk and numerous species will hybridize, ie. sika, red deer, sambar, rusa, and pere david to name a few.
Whitetail and mule deer will cross, but not readily, usually in forced scenarios, as behavior is different between the two, and cross breeding is limited.
Google, will produce much info on hybrids among animals with some being very interesting.
Red deer and whitetail do differ some genetically and it is unlikely that they will cross breed and produce live offspring. The old adage is never say never, as goat will breed shepp, but viable offspring usually don't occur. A couple of sheep/goat hybrids have supposed to have occurred, but this if this happened it was at a very low rate, much like the crossing of camels and llamas. A few years ago, we inseminated 5 elk cows with whitetail semen, with no offspring being produced.
Ray