The Shari'ah ruling is simple and straightforward that Zakaat cannot be given to non-Muslims.

Explanation
When the Munaafiqeen accused Rasulullah (Sallallaahu Alayhi Wasallam) of not being just in distributing Zakaat, Almighty Allah Ta'ala revealed the following verse which explicitly describes the recipients of Zakaat.

'As-Sadaqat (here means Zakaat) are only for the Fuqaraa (poor), and al-Masaakin (the poor) and those employed to collect (the funds); and to attract the hearts of those who have been inclined (towards Islam); and to free the captives; and for those in debt; and for Allah's Cause (i.e. Mujaahideen - those fighting in a Holy battle), and the wayfarer (a traveller who is cut off from everything); ........(At-Tawbah 60)

At the time when this verse was revealed the Prophet (Sallallaahu Alayhi Wasallam) commented, 'Allah did not leave determining the recepients of Zakaat to a Nabi or non-Nabi' (i.e. the Nabi of Allah himself determines who are the rightful owners of this money). (Ma'riful Qur'an)

Overtly, it may seem that this verse has no restriction in giving Zakaat to non-Muslims, but this verse should be understood in the light of the explanation given by Rasulullah (Sallallaahu Alayhi Wasallam).

Hadhrath Ibn Abbaas (Radhiyallaahu Anhu) reports that Rasulullah (Sallallaahu Alayhi Wasallam) sent Hadhrath Mu'az bin Jabal (Radhiyallaahu Anhu) to Yemen and ordered him, 'Call the people to bear witness that there is no One worthy of worship but Allah and verily I am the Prophet of Allah. If they obey you then tell them Allah has made compulsory upon you five Salaats during the day and night. If they obey you then inform them that Allah has made Sadaqah compulsory on their wealth - the wealth be taken from them (those who accepted Islam) and be distributed to their poor (Muslims).' (Bukhari vol. 187 p. 202)

'Awn bin Abi Juhaifa (Radhiyallaahu Anhu) reports from his father that the person appointed to collect Zakaat by Rasulullah (Sallallaaahu layhi Wasallam) came to us. He took Sadaqah from our rich and distributed it among our poor.' (Tirmidhi vol. 1 p. 82)

In the light of the above Ahaadith, it is clear that the recepients of Zakaat are Muslims only. The author of Tafsir Mazhari says that there is consensus of the scholars that Zakaat cannot be given to non-Muslims. (Tafseer Mazhari vol. 5 p. 308)

The author of Hidaaya, Allamah Marghinani says,

'It is not permissible to give Zakaat to a dhimmy (non-Muslim citizen in an Islamic state) because Rasulullah (Sallallaaahu layhi Wasallam) ordered Hadhrath Mu'adh bin Jabal (Radhiyallaahu Anhu) to take from their rich and give to their poor'. (Hidaayah vol. 1 Chapter on Zakaat)

Explanation to 'to soften their hearts'

In the understanding of many that these people were non-Muslims and Zakaat could be given to them. However, this too should be understood in the light of the practice of the Prophet (Sallallaahu layhi Wasallam). Allama Qurtubi has listed all the persons whom the Prophet had given Zakaat i.e. to soften their hearts. He emphatically states that all of these people were Muslims. (Ma'ariful Qur'an vol. 4 p. 401)

Therefore, in the light of the above explanation, it is categoric and clear that only Muslims are the recepients of Zakaat. This does not mean that Muslims cannot contribute to non-Muslims by means of other charitable funds. The prophet (Sallallaaahu layhi Wasallam) said,

'Give charity to people of all religions' (Hidaayah vol. 1 Chapter of Zakaat).

And Allah Ta'ala Knows Best.