No - you are domiciled at birth in the place (for an American a US State) that your father is domiciled in at birth. If you settle somewhere else permanently expecting to remain there for ever and abandon your conections to your previous domicile then you may acquire domicile in that new place.
Owning property, length of residence, nationality, pension rights etc does not by itself prove domicile. Rather one proves domicile by intention. So if you have settled in England & Wales, Scotland or Northern Ireland permanently then you are domiciled in one of these 3 countries. If alternatively you still call a State in the US home, keep your State drivers licence, vote there and may return there should your spouse pre-decease you then you would not yet have acquired domicile within the UK.
For most folk in the UK who can claim non-Uk domicile it is a money saving device because it enables one to put all of one's savings offshore and pay no UK tax on interest, dividends and capital gains unless these items of income are remitted to the UK.