As you undoubtedly realize, coming-out is an intensely personal and usually quite difficult process and there's often a certain amount of self-deception involved before you can even reach the point of telling others. I'd suggest not reaching out directly at all, but rather just be available and develop your relationship with your cousin however you normally would, setting all family rumors aside. He'll talk to you if/when he's ready and that's assuming he's even gay, which your friends and family may well be misjudging.
That said, you're going to be a better judge of your relationship with your cousin than any of us here. You can leave hints that you're willing to discuss sexuality without trying to imply anything and you can try to hang out with him in settings where he'll be free to confide if he wants to. Ultimately, though, he should make the first move. Direct confrontation may well just push him deeper into his shell.
One thing to consider here is how to interact with the friends and family members who are raising these concerns. If you're comfortable, share with them some of the difficulties you faced in the process and encourage them to give your cousin space to define himself however he will. Encourage them to consider that they might be wrong in their assumptions, as well. Granted, easier said than done, especially when it comes to family.