If I’m recommended for an MEB, can I fight it?

I assume you mean "referred to a MEB" and not "recommended", because it's not an option. You will have the ability to disagree with any findings that come from the Integrated Disability Evaluation System (IDES) (the MEB is the first part of the IDES), but you have no choice about being put in (well, you can plead with the person referring you, but that's about it).

There are a couple of routes the process can take, but in a nutshell it comes to this. At some point you will hit your Medical Retention Determination Point (MRDP). Most often this happens when you reach one year and still have a condition that prevents you from fully doing the functions of being a Soldier or your specific job (this most often happens when you are given a P3 or P4 profile).

One of two things can happen at this point - you will go to a MOS Administrative Retention Review (MAR2) or directly into the IDES process (first phase is the MEB). If you go the MAR2 route, the resultant decision will be to keep you in your MOS, reclassify you to another MOS, or referred to the IDES.

There are various phases in the IDES (MEB, PEB, etc) which you will be led through by the hand if you are referred. One of the very first things that will happen if you get into the IDES is being assigned a PEBLO (Physical Evaluation Board Liaison Officer) who, quite frankly, will serve as your spirit guide though the entire process.

In a nutshell, the process is to review and document fully any of the conditions affecting you and then determine what conditions cause you to be 'unfit for duty' (some conditions would make an 11B unfit but now a 92A).

Being referred to the IDES does not automatically mean that you will separate from the military as there are various "off ramp" points where you are returned to duty if you are not found unfit. During the IDES process, you'll have the opportunity to disagree with MEB or PEB findings.

Bottom line: If you're 'broke' and can't do your job, can you do another job in the military? If you can't then they will put you through the IDES and evaluate you for retention or separation.

I have seen a guy lose a leg at the knee and not only beat a MEB, but stay SF and deploy to Iraq again.
But he went into that board with an SF surgeon on his side, you won't have that.

That said, I wouldn't call the Army "stable" for a new baby, in fact it's the opposite of that.
If you love being a medic, you would probably love to work in an ER in any capacity; patient tech/CNA, LPN/LVN, RN, etc.
But if your injuries are permanent, you might also enjoy other career paths in the hospital outside of the ER that don't involve 12+ hours on your feet.
And many hospital systems have tuition and work/school programs to progress all the way from CNA to NP.
And the pay and benefits are better.

Combat Arms include: Infantry, Armor, Artillery (Field and Air Defense), Special Forces, Aviation (Air Cav and Attack Helicopters), and Combat Engineers. Medics are not considered Combat Arms. Can you be placed in Combat Arms units? Yes. But that does not make you Combat Arms.

Who said you cannot have a Perm profile? Is your Permanent Profile a P2 or a P3? A P2 will not trigger a MEB. A P3 or P4 will trigger one. Can you fight a MEB? Yes. As long as there is medical documentation stating that you are still, while perhaps limited, fit for duty/continued service. However, if you do have to go thru a MEB and the Final Medical Authority deems you are no longer fit for duty, there is nothing that can be done.