There were two major problems at the time, Reagan had public stated that the U.S. would not take sides in the disagreement, but did support the U.K. privately in the matter and second the feeling among the U.K. public was that the U.K. itself needed to handle the problem since the Falklands belong to them.
It was certainly considered preferable for British national pride, that they handled it ‘alone’ if at all possible - for that reason, the UK also declined offers of support from Australia, Canada, and New Zealand, all of which were ready to deploy ships and troops (I would guess a lot of smaller Commonwealth countries would have been willing as well). I remember a brief flurry to see if we could get our carrier HMAS Melbourne, ready to go as another carrier (answer - probably not due to a dockyard accident that hadn’t yet been dealt with - even without that, we’d have been surging a ship at the end of her life - she might have done the job but it would have been a serious stretch). But Britain decided to go it - mostly - alone.