Quite clear and unambiguously assur according to Rav Moshe, as he’s written in multiple published Teshuvos:
Igros Moshe OC I 113 page 177
“To offer one’s hand to a woman in the manner of those greeting others upon meeting. It is pashut that it is prohibited even for an unmarried woman since they are niddah and surely it is prohibited for a married woman”.
Igros Moshe EH IV 32.9 page 76
“To offer one’s hand to a woman in the manner of those greeting others upon meeting. It is definitely pashut that it is prohibited as I have written OC I 113. That is because one needs to be concerned for the issue
of derech chibah and taavah.
It is unusual for Rav Moshe to write in his teshuva, as he did here, that “It is definitely pashut that it is prohibited.” During Rav Moshe’s lifetime the modern attacked him for being an extremist. After his passing they’ve attempted to reinterpret history and claim Rav Moshe ruled the exact opposite of what he’s written.