No as far as i know, better forget it. Also, it seems that the support to copy PS3 downloaded Vita games to the console was also disabled after some system update some years ago, although you can still download the installation to the console (??). When you buy the game at PS3/PS App (again read the game legal throughly before clicking, i take no responsability) it will appear on the Download section of the Vita Store App (using the same account of course)...
I can give some more tips based on my own findings (EU region, i checked some months ago, so maybe there are changes):
- Basically the vita games you can play on ps3 are the cross-play/cross-buy games you will find on the store that hasnt been delisted, check some wiki for the list: Some i can confirm : Resogun, Helldivers, Alien Breed, Metal Slug3, Entwined, Spelunky,Dead Nation+Road to devastation, Counterspy, ... I think also these ones Oddworld new'n'tasty, The Swapper, Ape quest,deathmatch village,when vikings attack, Ecolibrium, Flower... Also, i can confirm that multiplayer PS3/Vita even with the same account works for most of what i tested...
- I think bought PSP games can also be downloaded directly on Vita, i prefer to check if the game is available by searching on the Vita Store: Manhunt 2, DOA Paradise, Thexder Neo (nice game BTW
),... A PSP game that ONLY appears at the PS3 store search is Sega Mega Drive Collection but downloads on Vita!. An I discovered that there was a previous PSP ZUMA game that might had been delisted...
- If im not mistaken, PSP DEMO "Loco Roco:Midnight carnival" is possible to acquire and download
- For PSOne games there is a real headache to search, i can confirm "Driver" for sure, and if the game is searchable on vita store will sure appear to download...
HOWEVER i have found some more EXCEPTIONS, if someone around might contribute to solve i will really appreciatte:
- Sega Mega Drive Collection and Darius Collection (PSP exclusives) only appears on PS3 search... Zuma and Cloudy with possibilities... seems delisted
- Megaman 10 and 11 were PS3 exclusives, but there is some cryptic message about "portable devices" that might led to mistake, it might be some typo, right? anyone?
- All PSP demos are available on the PS3 store but NONE will download on Vita directly, i found my way to download them on PSP and Chovy-sign to vita
Im sure you can connect other tutorials on the web with my findings, but feel free to ask if having doubts.
Just for the record, i have never been affilitiated to Sony or have any relationship with affiliates and such, ...