Yes. I've had two candidates in the past year whose background is in healthcare and they made the transition to consulting. One got an offer with a Big4, the other with a specialised consultancy.
The important thing in your case is to have a very well tailored application and to apply to the right thing. Let me take them in turn.
1. Tailored application. You need a cover letter and a CV that reflect well the transferable skills which you've developed in healthcare and which would make you an effective consultant. You should also highlight any distinctive performance you've had in the past few years to signal to the interviewer that you are a top candidate. Networking and referrals count a lot here too, but this requires a more detailed discussion that is outside the scope of this Q&A.
2. Applying to the right thing. You should do proper research to understand which would be your target firms and specifically roles within these firms. My advice would be to leverage your background to enter into the Healthcare practice of these firms (for the ones that have it) and to network before the interview with the Partners from the firms who work especially in the healthcare area. They are most likely to pull you in once they get a deeper understanding of your knowledge and skills. Both candidates that I mentioned above used this technique and it worked well for them.
Feel free to reach out directly if you have any follow-up questions. Happy to help.