Because he knows that if he tries to go after Frodo and Sam, the other two hobbits will be tortured and killed, and he refuses to countenance that:
'Our choice then,' said Gimli, 'is either to take the remaining boat and follow Frodo, or else to follow the Orcs on foot. There is little hope either way. We have already lost precious hours.'
'Let me think!' said Aragorn. 'And now may I make a right choice and change the evil fate of this unhappy day!' He stood silent for a moment. 'I will follow the Orcs,' he said at last. 'I would have guided Frodo to Mordor and gone with him to the end; but if I seek him now in the wilderness, I must abandon the captives to torment and death. My heart speaks clearly at last: the fate of the Bearer is in my hands no longer. The Company has played its part. Yet we that remain cannot forsake our companions while we have strength left.'
(The Lord of the Rings, Book III, Chapter 1, "The Departure of Boromir")
Aragorn certainly realizes that Frodo and Sam are in a hard place, and he does feel an obligation to aid them, considering that his ancestor arguably got the hobbits into this situation in the first place (as he says elsewhere, "it seemed fit that Isildur's heir should labour to repair Isildur's fault"1). But he is more certain of Merry and Pippin's torture and death than of Frodo and Sam's, and appears to have a gut feeling of sorts that it's his task now to protect the weak, rather than to accompany the Ring.
1The Lord of the Rings, Book II, chapter 2, "The Council of Elrond".