What would modern Inca look like?
Well, remember that had they driven back the Spanish, they would still have had smallpox to deal with, though attrition rates may not have been as high as in OTL, but 30% would not have been exceptional. The Inca would have lost nearly 2 million of his subjects. I imagine that this would have considerably undermined the claim of the king to divinity.
However, the Inca religion bears some interesting similarities to Catholicism - the belief in one God and his son, the concept of punishment or reward beyond death, and a lot of their ceremonies were noted for their startling resemblance to the Catholic mass. They had a religion heavily around ceremony and abstinence. I don't think it is a stretch of the imagination to see a future king, perhaps even Attahaulpa himself, converting.
The Inca calendar was still very much in development at the time of the conquest, so the Inca would have adopted the Roman calendar. They already had a complex system of laws, justice, military organisation and industrial relations that weren't incompatible with European ideals. Additionally, Attahaulpa made clear to Pizarro, I believe, his willingness to pay tribute to avoid bloodshed. With Pizarro defeated as suggested in previous post, it is not incomprehensible for the Inca to pledge their loyalty to Spain in return for being left alone. Also their architecture, communications and transport system were astounding. Access to European weapons would have confirmed the hold of the Inca and possible aided their further expansion.
In short, had Inca not been attacked again, and had reached some arrangement with Castile, it would have become an imitation of the European courts in the New World and, I believe, a strong Catholic state.