I think you severely underestimate how big the US is, how many deer there are, and how long this would take. Is it possible: yes. Will it take decades: likely.
I am not a veterinarian but there are things you can do, and that is being done (for swine flu for example), to prevent disease outbreaks in animals, also within wildlife. Primarily surveillance and control.
Swine flu is controlled within farms. Deer don't live in farms and if controlling/surveilling their population was possible, then other diseases such as lyme wouldn't be a thing.
I mean we have enough rednecks to get it done. Declare open season and even provide a bounty and it could be done shockingly quick. It's probably not a good idea though. I don't know, I'll yield to the science on this one.
The USA has many places without any roads at all, which require ~75 mile hikes to get to, like parts of Wyoming, and also shares a 2,5000 mile border with a neighboring country that is mostly uninhabited land. We do not have remotely enough resources to eradicate deer even if we were crazy enough to.
Not all hunters are rednecks. Also, clearly you have never gone hunting. Let's rephrase this in a more understandable way. Do you think it would be possible to kill/capture ALL the pigeons in even a single city