If by "spicy" you mean "hot", then yes, there was hot food in Asia long before the Columbian Exchange. Black pepper (Piper nigrum), long pepper (Piper longum), and cubeb pepper (Piper cubeba) are all native to southwest India, or more specifically, the Malabar Coast. There is archeological evidence of the culinary use of black pepper from north India at 2,000 BC. Genetic evidence of domestication shows it was domesticated in southwest India around 5,000 - 6,000 years ago.

From India, these peppers spread to southeast Asia and then to southern China, and influenced almost all eastern regional cuisines. There was export westwards too, but it didn’t catch on as much. The mummy of Ramses II (1200 BC) had peppercorns stuffed in his nose, Greeks received peppers from India as early as the 4th century BC, the Romans imported it by the ton. But it remained a rare and expensive spice in these areas, whereas in the east, it was quickly transplanted to islands across southeast Asia and became cheap and plentiful.

While black pepper arrived in Africa as early as 3,000 years ago, it didn’t seem to have spread beyond Egypt, and remained a rare and exotic spice even in Egypt. The current popularity of hot food in certain African cuisines is mostly due to capsicum peppers, which arrived after the Columbian Exchange.

The sensation of heat is due to capsaicin in chili peppers and piperine in black peppers. Both these chemicals act in an identical manner by stimulating the TRPV1 receptor, which is normally sensitive to temperature and pain. Because by definition the Scoville Scale is based on the dilution of capsaicin, it can’t be used to measure the heat of black pepper. But black pepper can be subjectively hotter than many varieties of capsicum peppers, and you can always increase the heat level in food by adding more pepper. So in terms of heat, many Indian cuisines were "hot" thousands of years before the Columbian Exchange, and influenced other cuisines across southeast Asia and southern China.

Of course, "spicy" also has a more generic meaning which isn’t heat-related, but rather to describe the taste of a variety of other spices. The mainstays of Indian and southeast Asian cooking: cumin, coriander, turmeric, mustard seed - have all been around in that region for thousands of years. So the food was spicy as well as hot long before the Columbian Exchange.