Depends on whether the self cutting is an indication of a disability or handicap. If so, and their decision was based upon your disability or handicap, then such would be discrimination based upon perception of handicap or handicap itself. However, a proper analysis of your case cannot easily be determined in this forum. I would suggest that you consult and experienced labor and employment attorney to assess your case.
Herbert Tan, Esq.
The National Newark Building
744 Broad Street, 16th Fl.
Newark, New Jersey 07102
(973) 735-2681 (W)
(973) 735-2682 (F)
305 Broadway, 14th Floor
New York, New York 10007
Member of National Employment Lawyers Association (NELA)
If the scars are not relevant to your job, then there's a good chance that the doctor will not even ask about them. If the doctor does ask, you should probably answer truthfully. If all the scars are old, it seems unlikely that the doctor would even inform your employer. If the scars are more recent and you're currently having issues with anxiety, this may be a good opportunity to obtain treatment.
As for your employment, your employer should not be making employment decisions based off of your scars. Since your self-cutting is likely to be associated with an anxiety disorder, doing so would likely constitute disability discrimination.
This information is provided for informational purposes only. This is not legal advice and this response does not create an attorney-client relationship. If you wish to obtain legal advice, you should retain and consult with a local attorney.