Which religion has killed the most people in its name, Christianity or Islam?

False Christianity bears more to blame for the bloodshed, since they claim to worship God. True Christians don't behave as such.

That amounts to hair-splitting. It hardly mattered to the victims.

All religions are false; world would have been better off if religion never got started at all in the first place. 
But, 'course it is in the nature of the human bang to create societies (religions) based upon any crazy little ideas that come along the way. 

It kinda is, militant atheism is one of the tenets of Communism due to a deliberate twisting of the words of Karl Marx.

Mao Zedong was never religious, his "great leap forward" was responsible for an estimated 45 million deaths, making him the worst mass murderer in human history. Incidentally, the natives of the Western Hemisphere were mostly annihilated some years before the Inquisition arrived - Columbus' ships carried rats, not all of them left with him. Bubonic plague wiped out between 70-90% of a people that had no acquired immunity.

Mao was indeed a prolific murderer, though many millions of his people died of collateral famine, not gunplay. But Mao’s total body count (uncertain as it is) may not measure up to the total amassed by the forces of Christianity (direct and collateral) over the many centuries. I would also argue that Mao doesn’t merit inclusion here, given the particular scope of the Q.

It wasn’t rats that slaughtered the Incas, Mayans and Aztecs, among others. It wasn’t rats that destroyed the cultures of the plains Indians, stole their lands, decimated their buffalo, brought smallpox, betrayed and enslaved them. White Christian men did these things. If we want to mention the rats, we must give due credit to those who brought them. It wasn’t atheists, Buddhists or Muslims. Columbus was a devout Christian, serving Christian benefactors, as were the rest of the Europeans who invaded the Western Hemisphere in the 15th, 16th and 17th centuries. It was also Christians who brought African slaves to the West, no count of those who died in this travesty is known, but likely more than a few. This post was edited by Don Barzini at October 26, 2018 11:52 PM MDT

Were those things done for the Christian cause or for simple greed?

We could say some of both. In the context of this question, it doesn’t matter, they weren’t done by Muslims. 

Columbus brought the plague-ridden rats inadvertently - so the disease that decimated the Native Americans cannot be said to have been "killing in the name of religion". It was accidental.
Those that survived were massacred, true. Had they not been severely weakened and thinned out by the plague, however, they may well have been able to push the invaders right back out again (as the Skraelings did to Erickson's Vikings, and the Maori almost did in New Zealand - England won New Zealand by trickery, not open warfare).

You may want to review the number of western indigenous peoples who died of plague. Compare that to the figures for those dispatched for alleged heresy and blasphemy simply because they didn’t abandon their own traditional beliefs. 

Had Columbus and his band of European invaders, as well as those who followed, -Christians all-  NOT come to the West, those rats wouldn’t have either. You say accidental, I say collateral. I point to your example of Mao as not individually or purposely killing every person, but causing mass death nonetheless. 

Columbus ”claimed” the lands he set foot upon in the name of the Spanish King AND (the Christian) God. As did Pizarro and Cortez. Religion was clearly in play, as it was in most matters of that era. This post was edited by Don Barzini at October 26, 2018 11:52 PM MDT

I would claim the major native civilizations of North America died out prior to Columbus expedition ... but likely still due to rats bringing plague.  The Anasazi and Mississippi "Mound Builders" both had large civilizations that ceased around 1600. 

While there is disputable evidence of pre-Columbus journeys by Knights Templars, there is no evidence they attempted to establish religious outposts or convert natives (let alone kill them for not being Christian). 

The Mi'kmaq people of Newfoundland have a legend about "white medicine men whose totem was a cross". Most likely inspired by Irish monks of St Brendan's order. They brought healing, not pain - but being monastic, the colony died out.
It's quite possible that Native American lacrosse was developed from Celtic hurling, the similarities between the two sports are striking - and as European wine grapes are not native to the Americas, sonebody must have brought them there before Erickson named the place "Great Vinland".

By that yardstick, true Islam has never engaged in bloodshed either. I've read the Qur'an and its companion volume, the Ashtiname - Mohammad exhorts the reader to "be good to the People of the Book (ie Christians and Jews). Invite them into your homes and dine with them."

"By that yardstick, true Islam has never engaged in bloodshed either"? I beg to differ. History has shown that both Islam and Christendom, (Not true christianity)are guilty of bloodshed since their existence.