The "sigh" is not necessarily a sigh of regret that he did not choose the other road, but a sigh of regret that he was unable to see where that other road would have taken him or where he would have been at the time he was writing the poem. There is a very subtle indication of a sigh in the poem itself, and it is one of the best things in the entire poem. When he says, "Two roads diverged in a wood, and I--I took the one less traveled by," that break indicated by a dash must be taken to represent the sigh itself. It would seem that he cannot tell about the event without feeling a strong emotion which causes him to stop in mid-sentence and sigh.
It is also interesting that he should say that he will be telling about this incident "Somewhere ages and ages hence." That sounds like hundreds of years in the future. Does this means he believes in personal immortality? It suggests that the road he chose to take a long time ago is an endless road, one that has already taken him a long distance but one he will have to continue traveling throughout eternity. That should be enough to make anyone sigh. It seems quite probable that he is not sighing because he regrets not taking that other road, or at least finding out where that other road led, but because he realizes he still has a long journey ahead of him, one that might take him forever. One might be reminded of the conclusion of Frost's most famous poem, "Stopping by Woods on a Snowy Evening," in which he repeats that he has "miles to go before I sleep / And miles to go before I sleep."
In "The Road Not Taken," the "sigh" in the last stanza is a sigh of regret. The speaker is telling this story with a sigh. He regrets that he could not have traveled both roads. The poem is about the other road--the road not taken.
Since way leads on to way, the reader doubts that he shall ever come back and take the road not taken. The speaker is sighing because he is saddened by the fact that he could not take both roads. He regrets not having decided upon the other road--the road he didn't take.
He realizes that he may be telling this story with a sigh of regret. Two roads diverged and he took the one road and left the other for another day. Still, he recognizes that he will probably never come back for way leads on to way. All the speaker can do now is imagine what the road not taken might have had to offer. Of course, he will never know for certain because he could not take both roads. In the end, he seems to think that the road he did choose has made all the difference in his life. Still he tells his story with a sigh:I shall be telling this with a sigh Somewhere ages and ages hence: Two roads diverged in a wood, and I— I took the one less traveled by, And that has made all the difference.
Whether that difference is for the better is something the reader will have to speculate upon.