They did colonize Asia. India was under British rule for over a century and most of Indochina was under French rule. Indonesia was a Dutch colony and the Philippines were Spanish. The areas of northern Asia colonized by Russia are so integrated into Russia that we don't think of them as colonies. Japan, China, Thailand, and some other Asian countries didn't come under European rule for a variety of reasons that deserve separate questions.
In conclusion, the answer to "why didn't they" is that they did.