Can the expectant father recover damages for emotional harm when medical malpractice causes a miscarriage or stillborn?

No.  In Broadnax v. Gonzalez, 2 N.Y.3d 148 (2004), overturning 19 years of precedent, the Court of Appeals ruled that two women who claimed medical error led to the deaths of their unborn children could recover for their emotional distress.  The facts in Broadnax involved the death of a full-term stillborn girl. 

The Court of Appeals in Broadnax limited a mother's recovery only to damages for the emotional distress attending a stillbirth or miscarriage caused by medical malpractice.  The court did not extend the holding to the father of the unborn child, reasoning that the physician does not owe a duty of care to the father of the unborn child.

The New York's Court of Appeals tailored the Broadnax decision to limit recovery to expectant mothers.  In footnote #3 of the decision, the Court of Appeals in Broadnax explicitly rejects emotional distress claims brought by expectant fathers.  The court ultimately decided that the duty of care runs only to the mother. Thus, the law only permits a mother to recover emotional damages if her unborn child does not have its own cause of action for pain and suffering.