He killed a Christian by mistake – should he fast for two consecutive months?

Whoever accidentally kills a Christian who is a dhimmi (one with whom we have a
contract or treaty of al-dhimmah (i.e., one who lives in a Muslim state)) or a musta’man (one who has entered the Muslim land and has been
guaranteed safety, such as those who come to do business, to work, to visit relatives, and so on) or is partly responsible for his death, must
offer expiation (kafaarah), according to the majority of scholars. 

Ibn Qudaamah  (may Allaah have mercy on him) said in al-Mughni (12/224): It
(expiation) must be offered for killing a kaafir who is protected (by sharee’ah), whether he was a dhimmi or musta’man. This is the view of the
majority of scholars). See also the answer to question no. 33683 

As the specialists have ruled that you are partly responsible for this accident,
so you are regarded as being partly responsible for this death. If a number of people are responsible for an accidental death, then each one of
them must offer expiation in full. This is the view of the four imams. See al-Mughni, 12/226. 

The expiation for accidental killing is to free a slave; if the person cannot do
that then he must fast for two consecutive months, because Allaah says (interpretation of the meaning): 

“and if he belonged to a people with whom
you have a treaty of mutual alliance, compensation (blood money — Diya) must be paid to his family, and a believing slave must be freed.
And whoso finds this (the penance of freeing a slave) beyond his means, he must fast for two consecutive months in order to seek repentance from
Allaah. And Allaah is Ever All-Knowing, All-Wise”

[al-Nisa’ 4:92] 

And Allaah knows best.