What is the likelihood of being able to enlist in the Navy after involuntarily separating from the Army?

The correct answer for any "is it possible to join the ______" question will always be "Talk to a ____ recruiter" as they are the only ones that will be able to give you the most accurate information.

To your specific question: Is it POSSIBLE? Yes. As has been stated, you will need a waiver.
What is the likelihood of it? You'll have to talk to the recruiter.

As others have pointed out, recruiting is a challenge for all the services and your chances of having a waiver approved are better than if you attempted just two years ago. There are some serious misconduct situations that will be rejected automatically (drug trafficking, most sex crimes, etc.), but as long as you don't have any of those and you didn't have two or more serious offenses, you should have a shot.

But only a Navy Recruiter will be able to give you the 'best' answer, because even they won't be able to give you any indications until they understand everything about your situation.

I'm sorry? What was that? You got the boot for committing a serious offense and you want to know if it's possible to re-join but doing so with the Navy? Your RE Code will need a Waiver. Your Sep Code will need a Waiver. To answer the question...yea, it's possible. Albeit a very SMALL chance. If you want a definitive answer then you need to speak with a Navy Recruiter. Now, if you HAVE spoken with a Navy Recruiter and they told you NO.....then hate to break it to you, Sparky...but that dog won't hunt.

You code means No Need To Apply, it isn't going to happen w/o a POTUS waiver- haven't ever heard of one. And your name isn't Biden