Can I negotiate for Ranger school when renewing my contract as an Army Dentist?

SPC Sohn, it sounds like you will be commissioned as a 63N, or perhaps 63A, if I read everything correctly. There are no commissioned officers under your branch (63) assigned to the 75th Ranger Regiment. That significantly weakens your prospects of attending Army Ranger School; the 75th probably could not fence you for an assignment. There are, however, unique situations and unique support assignments and you should always inquire. Do a search of the 75th Ranger Regiment and find their officer recruiting email address and reach out to them.
You cannot really "negotiate" your way to Army Ranger School but you can negotiate an assignment to a unit or Dental Health Activity that has Dentists/Surgeons assigned to support units or teams under JSOC/USASOC. If the unit has a requirement for a Ranger-qualified 63 (I only knew one 63N who was tabbed and in direct support to SOF) then your prospects just went up. You would need to attend Ranger Assessment, either RASP or maybe RASP2.
Don't forget your letters of recommendation from your boss and your boss's boss. If you explain to these people why you want to attend Army Ranger School and you describe it as you did above ... forget it. They too will see that your intentions are way off track. Best of Luck.

Mr. Sohn,
It really depends on the unit and leadership if they are willing to send you. Caveat: (I'm only speaking from experience here) You are fighting upstream here. If you are still in the reserves, it's only going to be three times as hard. The army is a people's game so you could play politics and get the right people in your corner to help you. These are the reasons why trying to go to ranger school will be hard:
1. You're support in every sense.
I don't know what it is about support MOS units but as soon as you want to do the cool stuff people want to fight against you. "Why do you need that?" "It's not in your career path." "You're just trying to be better than everybody!" Heard it all before. Higher echelon leadership will often thwart you out of envy or jealousy. I've seen this happen with 88Ms, 42As, and the like.
2. Mission planning.
Being that you're actually going to be commissioned as a dental officer, your role is going to be one of much responsibility which is not going to afford you the time to be able to go to places like Ranger school unless you have some form of replacement or good plan for when you're absent.
3. Unit funding.
If the unit cannot justify why send you the ranger school will benefit the unit, more than likely you will not go.
But with that being said, anything can happen so just do your best and put the bug into people's ears.

You can always ask but without there being a need or requirement it will be much more difficult to achieve. If you had a billet that required it or if you were in direct support of the 75th or a similar unit you would have more leverage but just keep pushing and keep asking. You might get lucky.