The French New Wave were all too different to generalise like this. Godard and Rohmer (or Resnais) could't be more different really.
It was actually the previous generation of French directors that they were all reacting against - who were making somewhat pretentious literary adaptation films. You can read about it in Truffaut's 1954 essay 'A certain tendency in the french cinema'.
Neorealism was actually a fairly minor influence - as none of the new wave directors were social realists. But older, maverick French directors like Bresson and Melville were much more influential, as it was their freewheeling sensibility which united all the new wave directors and sent them hurtling off in different directions.
Yeah, it's hard to talk about French New Wave as a cohesive set, since it was so reactionary - I still see influences from earlier italian neorealism in Godard, in the street view slice of life shots and focus on regular people throughout Breathless
Not sure I agree De Sica's Sciuscià (Shoeshine) won a Honorary Academy Award in 1947, which then became the Academy Award for the Best Foreign Language Film. In other words it was so influential that it ended up defining its own category at the Oscars.
I am wrong about that then, but I was talking more about influence on American films - I haven't seen any Sciuscia so I don't know if that went on to influence american films in the same way that godard did later.
Edit: I had originally prefaced this with "I'm not a scholar" but forgot to readd it - the responses below invalidate most of what I said, but leaving the comment so their responses make sense - though I still stand by the lesser version of what I said - that Godard had a bigger impact on American cinema than pre-Fellini Italian Neorealism, and that parts of italian film influenced Godard.