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After all, Frodo's mission is more important, is it not, than saving Merry and Pippin.

IMO, it is precisely because Frodo's mission was so vital that Aragorn made the decision he did. It was an act of faith on his part. He reasoned that Frodo's destiny was not in his hands because more capable hands were guilding him, not the least Frodo's own will. Remember the words of Gandalf to Frodo about how Bilbo found the Ring, those forces (or that force) was still active, and I think it was that realization, coupled with Frodo's own actions, that led Aragorn to decide that Frodo had passed beyond his aid.

Once Aragorn decided that Frodo was beyond his aid, going after Merry and Pippin was a simple choice to make.

Note that he said "My heart speaks clearly". This means, to me, that is decision was an act of faith, rather than simple logic. This has much to do with things like 'foresight' and assorted other gut reactions that occur in fairly regularly in M-e.

So, I think Aragorn saught to decide with his heart rather than his head, and, being the fine upstanding guy he is, his heart told him true, and he made the right choice in a bad situation.

*Spoiler*

As an aside, I will note that even if Aragorn had decided to go after Frodo, Rohan still would've been saved to help Gondor as Gandalf still would've shown up to confront Wormtongue and heal Theoden. All would've been more or less the same *except* that Aragorn would not have taken the Paths of the Dead and Gondor, ultimately, would probably have fallen at the Battle of Pellenor Feilds.

Tuor

All would've been more or less the same *except*

And perhaps *except* Frodo's mission (and its outcome?) would have been altered. Certainly, Aragorn could not have made it to the Mountain of Doom in the manner which Frodo and Sam did. Nor would the capture of Frodo been treated as cavalierly by the Orcs had Aragorn been present (either he or Sam would have been found as well, since only one could wear the ring). His presence could have caused the ring-bearer's mission to go awry in a number of ways.

Maybe that was part of what his "heart" spoke so clearly.

..This means, to me, that is decision was an act of faith, rather than simple logic. This has much to do with things like 'foresight' and assorted other gut reactions that occur in fairly regularly in M-e.</>

Exactly! Tuor, also his respect for others 'free will'.
Boromir's error was in NOT looking to faith NOR respecting others choices and abilities. Boromir relied on what he could see physically. That's why he seemed to patronize the little Hobbits and did not respect their internal strength.