I do not know of any reason why one cannot do that, because Allah, may He be glorified and exalted, says in His Holy Book (interpretation of the meaning):

“Allaah does not forbid you to deal justly and kindly with those who fought not against you on account of religion nor drove you out of your homes. Verily, Allaah loves those who deal with equity”

[al-Mumtahinah 60:8]. 

So Allah is telling us that with regard to those disbelievers who do not fight us or drive us out of our homes, He does not forbid us to be kind to them, treat them nicely and help the one who is in urgent need of help. The mother of Asma’ bint Abi Bakr (may Allah be pleased with her) came when she was still a disbeliever to her daughter in Madeenah, at the time of the truce between the Prophet (blessings and peace of Allah be upon him) and the people of Makkah, to ask her to uphold ties of kinship. Asma’ consulted the Prophet (blessings and peace of Allah be upon him) about that and he told her to uphold ties of kinship with her, and said: “Uphold ties of kinship with your mother” when she was still a disbeliever. 

So if a non-Muslim who is living under Muslim rule or a non-Muslim who is on peaceful terms with us, with no war between him and us, is in need of that, there is nothing wrong with giving blood to him as an act of charity, and you will be rewarded for that, because there is nothing wrong with giving aid to one who is in need of charity. End quote. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him).