Yes, you can. But may you? According to Emily Post’s original Etiquette, published in 1922: yes, with reservations.
Although asparagus may be taken in the fingers, don’t take a long drooping stalk, hold it up in the air and catch the end of it in your mouth like a fish. When the stalks are thin, it is best to cut them in half with the fork, eating the tips like all fork food; the ends may then be taken in the fingers and eaten without a dripping fountain effect! Don’t squeeze the stalks, or hold your hand below the end and let the juice run down your arm.
Later versions of Etiquette uphold these rules but even less enthusiastically.
By reputation this is a finger food, but the ungraceful sight of seeing a bent stalk of asparagus dripping like a fountain into someone’s mouth, and the fact that water is also likely to drip from the end, has been the reason why most fastidious people invariably eat it — at least partially — with a fork. That is, cut the stalks with a fork to where they become harder, and then pick up the ends in the fingers if you choose. But don’t squeeze the stalks, or hold your hand below the end and let the juice run down your arm.
More recently, the Emily Post Institute gave a tepid, qualified endorsement of fingers:
While asparagus may be eaten with the fingers, this only is if the spear is crisp and has no sauce or butter on it. However, if everything else is being eaten with a fork and a knife, it is far preferable to use cutlery to eat the asparagus, too, even if it is unadorned.
So basically, you can eat asparagus with your fingers, and we do. But be forewarned: you're edging up to the very borders of acceptable behavior.