Is it somehow possible to use the newtxtext and newtxmath fonts, and corresponding .sty files — as-is or modified (and if the latter, how) — with XeLaTeX instead of pdfLaTeX?
I note that:
Yes, you can write \usepackage{newtxtext, newtxmath} and compile with XeLaTeX. You might get an obscure bug related to Greek letters in \mathrm. (ETA: Commenters bring up more bugs with hyphenation patterns and non-ASCII characters than I was aware of.)
You can also use the font TeX Gyre Termes Math in unicode-math, which has a very similar appearance to newtxmath, but all the modern features and a larger selection of glyphs. There is also a text font, TeX Gyre Termes X, specifically based on the tweaks to Times in txfonts. Or you could go halfway, using an OpenType version of Times with fontspec along with math from newtxmath.
Also note that new development of TeX is currently focused on LuaTeX. There are a few packages that still only work in XeTeX, but of the features I need, HarfBuzz font rendering was finally added to LuaTeX this year, ucharclasses doesn’t work, and I prefer unicode-math to mathspec. There are still a few miscellaneous packages I might need XeTeX for here and there, but I’ve been using LuaTeX when I can for years, due to its better support for microtype.
Since it’s better to use Unicode fonts in Unicode engines, use iftex to detect what kind of system you’re compiling on, if you must support both.
\usepackage{iftex}
\ifTUTeX
\usepackage{unicode-math}
\defaultfontfeatures{ Scale=MatchLowercase, Ligatures=TeX }
\setmainfont{TeX Gyre Termes}[Scale=1.0]
\setmathfont{TeX Gyre Termes Math}
\else
\usepackage{amsmath, newtxtext, newtxmath}
\fi
There should be very few incompatibilities. One that I’ve seen is capital Greek letters such as Δ within \mathrm, which happen to work in newtxmath. In unicode-math, you would use \symup\Delta or \upDelta instead.
If you want newtxmath instead of unicode-math for some reason, alternatives are to load fontspec with newtxmath and TeX Gyre Termes as the main font, mathastext, and mathspec, which also gives you letters from an OpenType font in math mode. If it’s important that the XeTeX/LuaTeX output be as identical to the PDFTeX output as possible, use TeX Gyre Termes X as the main font.
newtxtext, but some other font such as tempora?
\documentclass{article} \usepackage{newtxtext, newtxmath} \begin{document} abc \end{document} you will get no error but no hyphenation patterns are loaded for 8bit fonts so hyphenation if it works will just be accidental mis-interpretation of the utf8 patterns.
\documentclass{article} \usepackage{newtxtext, newtxmath} \begin{document} Grüße \end{document} with XeLaTeX results in "GrüSSe" instead of "Grüße" as expected, so 8bit fonts in Unicode engines are not only dangerous because of hyphenation patterns. (Same with LuaLaTeX.)
\iftex: I can compile in either pdfLaTeX or XeLaTeX (or LuaLaTeX). Which I use will be determined by what choice I make in text and math fonts — whether they are completely (or nearly completely) unicode-based.