I remember reading somewhere that in OTL, Cabot probably thought that he had reached a northeastern extension of Asia when he actually found either Newfoundland or Nova Scotia. In 1497, many still thought that Columbus had discovered large islands between Europe and Asia, and that sailing further west would lead to Asia, which was still no more than 4000 miles or so west of Europe by sea. If there was a large northeastern extension of Asia (like the real life Kamchatka Peninsula, but bigger), navigators thought that it might be located in the same place as Newfoundland and Nova Scotia were actually located.
If Cabot is the first to find North America, I suspect he would still believe this. Therefore, he would try to sail down the east coast of OTL Canada and the United, believing that he was in northeastern Asia and that Cathay/China could be reached going down the coast. If he can reach Florida, it will be obvious that this is not Asia, but in an attempt to reach Asia, Cabot or another explorer will probably push west into the Gulf, and eventually reach Mexico, where they will make contact with the Mexica. This is assuming that the English king and merchants are willing to fund further expeditions after the first couple make contact only with natives who live in small towns and do not possess any of the luxury goods that would excite investors of the time. If the explorations stop before they reach the richer civilizations of Mexico and Central America, those civilizations might have another century, or at least several decades, before European contact.
On the other hand, even if he never gets south of Florida, Cabot will stake an English claim to everything from Newfoundland to Florida, which may well be followed up by other explorers within a generation, hoping to find a passage or riches that Cabot missed. Exploration thrived on optimism - lots of explorers and investors in the sixteenth and seventeenth and into the eighteenth centuries were convinced that they could succeed in making discoveries that everyone else had missed.