Can I apply for postdoctoral fellowship in H4 visa?

I am currently residing in the US in H4 visa as a dependent of my husband (H1B visa). I have recently completed PhD from India in biomedical science and would like to pursue my career in the US as postdoctoral fellow. Can I apply for the different postdoctoral positions in my current H4 visa status? What is the exact procedure for availing the post doctoral fellowship?

You can apply but to accept a position will require you to have active employment authorization. This can either be done with an H-4 EAD (if eligible per USCIS rules) or you can request University to process your H-1B which is usually cap exempt. I suggest not discussing the H-4 and H-1B situation until after a job offer has been made by a Program Chairperson or Head, most Professors don’t know what a specific immigration situation means and there is no point in confusing them. Remove all immigration references from your resume. If applying online answer ‘yes’ if asked ‘are you eligible to work in the US?’ and state if possible in comments ‘With USCIS permission’.