Military medical malpractice claims are different than state law claims. A claim against a military hospital (like the VA hospitals around the country) is governed by the Federal Tort Claims Act (the FTCA). This cause of action however, does not exist for an active duty military person.
Active duty military men are prohibited under a case called Feres from suing. The Feres doctrine specifically precludes claims against the military by active duty personnel. However, it does not preclude a case brought by dependents of the active duty person. If you are not active duty, you may bring a claim under the FTCA.
If you bring a claim, you are not denied rights to receive continued medical care at a VA facility. In fact, in many FTCA cases, the government does not pay for future medical care at a private hospital since you would still be eligible for all future medical care at a VA facility.
FTCA claims require that an administrative claim be initiated before a lawsuit can proceed. There is a specific claim form that must be filed out to initiate that administrative process. The administrative claim must be given 180 days for investigation and determination. If you have not received a decision within 180 days, then you have the right to request what is called a "right to sue" letter and proceed with the filing of a lawsuit in Federal Court.
These cases can be tricky, so I would advise you to contact an experienced medical malpractice attorney who has handled military medical malpractice cases to determine whether you can sue under the facts of your case.
If you were not active duty military when this occurred, you can potentially bring a case, which starts by filing a "standard form 95" with the hospital. My website page on military malpractice has a link to this form: http://www.patrickmalonelaw.com/lawyer-attorney-1288740.html. But you should hire a lawyer experienced in FTCA cases. To answer your headline question, yes, filing a claim doesn't make you ineligible for services in other departments, but just remember they are human beings and some of them may be reluctant to treat you. I think being upfront and candid with the other providers about why you're pursuing the claim is the best remedy.