The simple answer is no.
The 9000 is backward compatible with 7900, and with that the relevant compatibility chart is
here.
The newer Shimano sti's pull more cable, around 20% more. The key implication mechanically is that new levers on old calipers will give you weaker brakes, while old levers on new calipers will give you stronger brake force but the levers will bottom out more readily than expected if you don't have true rims and tight pad/rim clearance.
The complicated answer, is that since the difference is only 20% which can easily manifest in different makes/models/reach in brakes and levers, the incompatibility is generally considered to be manageable - you just have to take a bit more care in setup/test and pad adjustment/choice etc. to be safe.
Hope it helps.