I'd like to point out that the Genoese were very involved in the Spanish empire. The Canaries, where Castille learned most about colonization were funded mostly by Genoese and Flemish sources, these were the men that brought the capital, knowledge, connections, and skilled workers required to make, refine, and ultimately sell the sugar from the plantations. There was in many ways that Genoa influenced the Castilian new world, by the 16th century where a handful of Genoese bankers welded immense power within and profited generously from the Hapsburgs. I don't think it mattered that the Spanish colonies was Spanish in name if it benefited the Genoese more.
So, can we change some things around, so that 1) Iberia is more divided with a big part of it, preferably including the region around Gibraltar, is downright subordinated to a strong Genoese ruling clique; 2) in its home region, Genoa is strong enough to repel all comers, be they imperial, papal, French or Ottoman 3) Genoa is motivated to look west for opportunity--say they are very strong in the western Med but ruthlessly excluded from the Eastern Med.
Suppose an early Portuguese-Genoan alliance breaks apart, with the Genoans having shared in Portuguese advances south along Africa and picking up all manner of Portuguese seacraft, including constructing ships meant for Atlantic operations in their southern Iberian colony. (How do the Genoans control the region of Gibraltar and Grenada? Well, suppose that some Iberian allies invited the Genoese to take Granada if they could, to take pressure off their frontiers, long before the Iberian powers were in a position to do it all by themselves? This may have been centuries earlier. The Genoese expand their trade with the Atlantic seaboard of Europe having secured Gibraltar for themselves--it doesn't enable them to shut the straits to all others but it makes it very very easy for Genoese and other shipping they favor to be sure of passing the straits with minimal grief--especially if the G's then secure the south side of the strait as well by sheer conquest.) With a political shift, the Portuguese turn hostile to the Genoese and manage to shut them out of the coastwise African route, which makes people who think like Columbus more interesting to the Genoese. Even a scholar who avoided Columbus's two egregious errors (underestimating the radius of the Earth and overestimating the east-west range of Eurasia, which combined would place Japan and China around the longitude of the North American coast) it would still be possible to argue that there is likely to be some land on the vast great circle route from Seville to China; perhaps Columbus himself or a close ATL expy might be persuaded the positive evidence he collected from Northern Europeans and Basques that there was something in range to the west did not prove it was in fact China, but anyway the notion that sailing due west should achieve some landfall that could be colonized as a way station to China is well founded. Holding Seville, or even just Gibraltar and a thin strip of countryside to back it and being originally based out of Genoa itself, and with some holdings in Macronesia as well, the payoff of America could work as well for a Genoese centered empire as for one centered in Spain. Anything they suffer in distraction due to holding a less secure base (unified Spain, even minus Portugal, is pretty well defensible at the Pyrenees after all) and perhaps a smaller core population they might make up for with shrewder administration, a greater willingness to partner with allies and recruit from diverse populations for their ventures; the upshot might be that Genoese Italian is just one language among many in a realm as vast as OTL Spain's Empire of the Indies but maybe the regime founded is more durable and stable; come 1850 or so maybe the Genoese West Indies are still formally under Genoese hegemony albeit becoming the tail that wags the European dog, or maybe a la Brazil OTL the regime shifts its capital west to say the Caribbean, a Caribbean from which they have more successfully excluded Atlantic European powers whose colonies were driven to the marginal region of the north American Atlantic coast, and not all of that--say the Genoean power even has a strong hold on the Chesapeake Bay and lower Mississippi as well. The notion they can do better than Castile is based on an assumption the Genoese core elites are more willing to share power with suitably checked and balanced partners, so in the vast Genoese empire there are swathes of settler or merchantile/plantation colonies dominated by clusters of other ethnicities, and maybe more synergy with coopted/elevated Native American protectorates filling in the gaps, but overall the whole thing has been cleverly arranged so self-interest of dominant classes keeps them loyal to the central power--maybe in the course of that center shifting from Genoa to the viceregal administrative center in America (presumably with a quite different name, though maybe by some counter-butterfly effect Amerigo Vespucci is in an even stronger position to get the place named after himself) Genoa itself remains affiliated, gracefully accepting a peripheral but profitable and well-supported position within the whole; now "Genoa" is basically a long-unified region around the entire western Med, including a long-subjugated (with difficulty) North African coast east to Tunisia and west past the strait, the entire Mediterranean coast of Iberia, as well as having taken all of the former French Med, the islands of the western Med, and Italy's west coast down to Naples. This region being under Genoese hegemony so long it is about as unified culturally and politically (except for the Muslim-majority north African fringe) as France by that date OTL; the stubbornness with which South Italy and Sicily resist incorporation is the stuff of interim history. ITTL "Italian Unity" is a matter of either submitting to Genoan imperialism or a vision that excludes the northwest coast of what we view as Italy from the peninsula on some grounds or other.
Could someone build at least part of this up as a solid ATL? The notion that the Genoese realm in the Western Hemisphere should be larger and more enduring than the Castilian is a bit of romantic hyperbole on my part to be sure, but I think it could plausibly be at least as large and at least as durable.