That’s not the case. It’s still Pekin or Peking in French, German, Russian, Polish, Swedish, Italian (Pechino), Finnish, Serbian, Croatian, Dutch, Greek, Spanish, Hungarian - the list goes on. See Wikipedia. We should beware of thinking everyone follows English-speakers.
There’s quite few places named for Biblical persons or places. Jerusalem was mentioned above as were some others. Amman Jordan was called Philadelphia in the Bible, and that may have influenced Penn’s naming decision.
More obscure places: Mount Pisgah and Mount Tabor are Biblical names. There’s several places named after each, including an extinct volcano within the city of Portland. The name Ai is also Biblical, being the name of some Canaanite city. There’s a few obscure places in the US with that name.
I’m sure there are many more.
And there’s always Intercourse, Pennsylvania. Can’t think of what that place is named for.
It’s loosely derived from “crossroads”, or “intersection”. Which is why the route from Blue Ball to Paradise really is through Intercourse. 
There’s a Mecca and a Jasmin in California.
I live in Macedonia, Ohio, which is named for the biblical Greek region and not the current Republic.
Are there many Eastern European place names in America or was everything already named by the time the Poles and Slovaks got here?
Krakow, Zachow, and Pulaski are three small burgs in Wisconsin not far from Green Bay.
I dunno, it worked quite well forElvis Costello.
Panama City, Florida is named after Panama, which is an indigenous word meaning “abundance of fish” (or maybe butterflies).
Quito, Mississippi, is named after Quito, Ecuador, which takes its name from an indigenous group.
Lima, Ohio, is named after Lima, Peru, which takes its name from a local oracle in an indigenous language.
York has always been an important place, if not necessarily a large one. Four Roman emperors ruled from here. The Archbishop of York has a palace here. The Viking kings of the Danelaw ruled from here too, not to mention the Council of the North (and the Railway King).
But New York was named after the man, not the town; the Duke of York is the title given to the second oldest Royal Prince, in this case the somewhat disappointing future James II.
One can only imagine why Cut and Shoot, Texas is named that way. And they would be pretty close to being right.
There are a lot of places named for classical cities in Greece (and in Carthage, which counts as non-European) in upstate New York - Attica, Syracuse, Rome, Utica, Troy (which was in the Middle East)
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Greece is in Europe. The OP asked about non-European names.
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Macedonia is not a biblical name. I think you meant “classical”.
There’s a fair number of eastern European names: Warsaw, IN; Moscow, ID; Belgrade, MT; Prague, OK; etc. But you know, they’re European too.
See #1 above.
#1 doesn’t mention a single one of those cities.
“We are marching to Peoria…”
Several members of my class preferred to sing it this way when I was a kid in Chicago. It is possible that I was among them. Old marching song from the Boer War, my foot.
York had to be a place name since it was described as “new”. The Duke of York couldn’t be “new”.
Considering Peoria’s metaphorical genericness, perhaps it was a marching song from the Bore war?
Truth or Consequences, NM changed its name from Hot Springs, in order to win the honor of hosting the popular radio program.
Many of the foreign city names transferred to US cities have undergone changes in pronunciation:
-Russiaville, IN is pronounced Roosha-ville.
-Cairo, IL comes out Kay-roe.
-Valparaiso, IN is Val-pa-RAYS-oh, different from the Chilean city.
I meant #1 in the list in my post. You know where I say that Greece is in Europe and the OP is not asking for European cities.
There’s Delhi, NY. Mexico, NY and Andes, NY take their names from places in Latin America with indigenous names (Mexica and Quechua)
I thought that Batavia, NY might also fit the bill, since Batavia is the old name for Jakarta, but it’s named for another Batavia: it’s Latin for the Betuwe region of the Netherlands.
There are also a number of towns named Alexandria; the one in Virginia is the biggest one I know.