The reason why the exception was made for Bilbo and Frodo does not apply to Aragorn.
Bilbo and Frodo were allowed to sail West in order to obtain healing from the effects of the Ring before they died; this is made clear in several of Tolkien's Letters, and I'll cite Letter 246 as an example:
'Alas! there are some wounds that cannot be wholly cured', said Gandalf – not in Middle-earth. Frodo was sent or allowed to pass over Sea to heal him – if that could be done, before he died. He would have eventually to 'pass away': no mortal could, or can, abide for ever on earth, or within Time.
Aragorn obviously did not suffer the effects of carrying the Ring, and so he does not need this healing.
There is, however, another reason, and that reason is that Aragorn's Kingship of Gondor and Arnor was part of Ilúvatar's plan since the Creation: the beginning of a time known as the Dominion of Men:
And some have said that the vision ceased ere the fulfilment of the Dominion of Men and the fading of the Firstborn; wherefore, though the Music is over all, the Valar have not seen as with sight the Later Ages or the ending of the World. (Ainulindale)
... in either chance the powers of the Three must then fail and all things maintained by them must fade, and so the Elves should pass into the twilight and the Dominion of Men begin. (Of the Rings of Power and the Third Age)
But the Dominion of Men was preparing and all things were changing, until at last the Dark Lord arose in Mirkwood again. (Of the Rings of Power and the Third Age)
That this can be explicitly equated with Aragorn's Kingship is seen in Gandalf's words to him on Mount Mindolluin (Return of the King, The Steward and the King):
This is your realm, and the heart of the greater realm that shall be. The Third Age of the world is ended, and the new age is begun; and it is your task to order its beginning and to preserve what may be preserved. For though much has been saved, much must now pass away; and the power of the Three Rings also is ended. And all the lands that you see, and those that lie round about them, shall be dwellings of Men. For the time comes of the Dominion of Men, and the Elder Kindred shall fade or depart.
Aragorn's union with Arwen also served to reuinite the lines of the Half-elven, as we read in Return of the King Appendix A:
There were three unions of the Eldar and the Edain: Lúthien and Beren; Idril and Tuor; Arwen and Aragorn. By the last the long-sundered branches of the Half-elven were reunited and their line was restored.
All of this is therefore part of the Creator's design for the world. Gandalf (who we would expect should know better in any case) certainly would not actively work against that design.