I am looking at Wikipeda for the proof of Chow's Lemma. My impression is that the proof uses $X$ is a separated scheme over a Noetherian scheme $S$ (throughout where it is argued that some graph map is a closed immersion we would only need $X$ is separated over $S$). At first glance I do not see where $X$ is separated over $S$ is not sufficient to use this proof. Could someone point out to me where "proper" is necessary and separated is not sufficient? I wouldn't dare ask for an example where the Lemma is not true if $X$ were just separated over $S$. But if you feel ambitious ... :)
Edit: It seems to me that the proof in Wikipedia is not very precise. I think I know what's going on now. We just need the fact that $X$ is finite type over $S$ (but proper is not really needed), at least this is the impression I get when I looked at the Stacks Project
The proof in Wikipedia is awfully written at the end (what is $v$?, what is $U′$?) and the problem is in this last part. Using the notation of Wikipedia, you can prove (without properness of $X$) that the map $g:X'\rightarrow P$ is a locally closed immersion. Hence you get Chow lemma without the properness hypothesis but only with a quasi-projective domain in the birrational surjective map.
But if moreover $X$ is proper then $$q_1:X\times_S P\rightarrow P$$ is a closed map, hence $g=q_1|_{X'}$ is a closed immersion and you get that $X'$ is projective. In order to see the entire proof written carefully look at the EGA reference provided there ( EGA II Theorem 5.6.1).