This is unclear.
Unfinished Tales implies that Aragorn did claim it for himself (bold is my emphasis, italic is in-text):
It must however be noted with regard to the narrative of The Lord of the Rings that over and above such deputed authority, even hereditary, any "heir of Elendil" (that is, a recognized descendant occupying a throne or lordship in the Númenórean realms by virtue of this descent) had the right to use any of the palantíri. Aragorn thus claimed the right to take the Orthanc-stone into his possession, since it was now, for the time being, without owner or warden; and also because he was de jure the rightful King of both Gondor and Arnor, and could, if he willed, for just cause withdraw all previous grants to himself.
Unfinished Tales Part 4 Chapter III: The Palantíri
However, it's not clear whether this refers to Aragorn claiming the Orthanc-stone in The Two Towers, rather than the line under discussion:
'There is one who may claim it by right. For this assuredly is the palantír of Orthanc from the treasury of Elendil, set here by the Kings of Gondor. Now my hour draws near. I will take it.'
Gandalf looked at Aragorn, and then, to the surprise of the others, he lifted the covered Stone, and bowed as he presented it.
'Receive it, lord!' he said: 'in earnest of other things that shall be given back. But if I may counsel you in the use of your own, do not use it - yet! Be wary!'
The Two Towers Book III Chapter XI: "The Palantír"
Unfortunately, it doesn't appear as though Tolkien ever resolved the ambiguity (or, indeed, if he ever considered it ambiguous); there's nothing I can find in Letters or History of Middle-earth that comes down one way or the other.
