I don't know if this is a match, but there is hope. On my great grandmother's marriage certificate (James' sister) her father is listed as "Edward Falvey". The names James and Edward are in your family also.
I have been helped in my search by James' middle initial V. The 1871 English census lists a James V. Falvey, age 29, born in Ireland, occupation Law Stationer. His wife Annie was 27 y/o. (I have no idea if this is my relative but am not discounting it yet.)
James V. Falvey petitioned for naturalization in New York in 1882; occupation Lithographer, witness Alfred Westbrook. The 1890 New York City Directory lists Jas. V. Falvey, engrosser. Home addresses for him are in the East 100's.
James V.'s death certificate (1906, New York City) lists his mother as Hanah (sic) Walsh. If the James V. from the 1871 census is the same person, then Rose C. Falvey was a second/subsequent wife. The 1910 US census shows Rose was born around 1857 in Louisiana (French father, German mother). Rose died in 1927. Both James V. and Rose are buried in St. Raymond's Cemetery.