An article authored by some Centers for Disease Control personnel suggests that such a procedure might exist in general (without ref. to New York City particulars):
if the family opposes a suicide determination and the ME/C finds the evidence supporting suicide lacking in any way, the ME/C may declare the death undetermined to avoid a potential legal dispute
(ME/C - there stands for "medical examiners and coroners". The reason for the latter distinction is somewhat complicated, but not terribly relevant here since New York City has abolished coroners in 1918, according to Wikipedia, although other New York [State] counties might not have done that.)
So, how can one legally dispute that finding of suicide recorded on the death certificate in New York City, in particular?