While there is a way to get the game to work with a modern resolution unmodded by messing with the config file, I do not recommend it because it will cause crashes at various points throughout the campaign; usually when loading a new level such as at the end of Tram mission (level 3).

There are, however, two excellent ways to play RTCW with native resolution support - the ioRTCW source port or the RealRTCW mod. The option you pick depends on how important a vanilla experience is for you, with RealRTCW being a faithful mod with minor gameplay changes and ioRTCW being 100% vanilla. I'll post instructions for installing both below:

Option A) For starters, I would recommend giving RealRTCW a try on Steam. It's an excellent fan-made remaster that's very faithful to the original game, heavily customizable, and allows you to play at a modern widescreen resolution. It also has achievements and workshop content. If you play and complete the game through this mod, no one's gonna say you didn't beat the game as it doesn't change the level design or the core gameplay. The only changes from vanilla RTCW are some minor balance tweaks and brand new weapons added (which you can turn off in the Game Options by switching the weapon arsenal to Classic). And, for those who want as close to a vanilla experience as possible, there's also the option of downloading the Vanilla Plus DLC for the mod, which mainly adds back the default weapon models, sounds, and textures while keeping the quality of life improvements like widescreen support. Here's the link:
https://store.steampowered.com/app/1379630/RealRTCW/

Option B) However, if you're a total purist and want a 100% vanilla experience in its original form, then you're instead gonna want to download the source port for the game called ioRTCW.
You can basically think of ioRTCW as the GZDoom or Quakespasm equivalent for RTCW. This sourceport will also run even if your current Steam installation of RTCW doesn't. It will give you widescreen and native resolution support in the in-game options as well as a few other quality of life features, but absolutely zero gameplay changes.
https://github.com/iortcw/iortcw/releases/tag/1.51c
Instructions: Download the correct zip file for your system from the Github link above. For example, if your system is a 64-bit Windows, then download "iortcw-1.51c-win-x64.zip". Using a program like WinRAR or 7zip, extract the files into your C:\Program Files (x86)\Steam\steamapps\common\Return to Castle Wolfenstein folder. When it asks you if you want to overwrite a readme file, just say Yes. Here is how your files should look in your Return to Castle Wolfenstein folder if you did everything right:
https://imgur.com/a/gMN3XdR
From there, you can simply doubleclick the "ioWolfSP" executable to open the game in the source port.

I recommend RealRTCW since it does everything ioRTCW does but better. Whichever option you do choose, when you boot it up, go into the in-game options and select "Automatic (Native)" from the list of resolutions. It'll automatically adjust to your native monitor resolution of 2560x1600. Hope that helps!

Thank you for the in-depth comment!

One thing though... how would you fix it with the config file?
I played Quake a lot times with a source port and, as close as it was to the original, when I tried playing it for a little bit I regretted not having played it that way in the first place because it looked a lot grainier and felt a little bit different too in the movements.

So that's why I'd rather stick to the original thing this time and just fix the resolution that way. About the crashes, I saw on a youtube video that if you just change the resolution and then put the native one back, you can pass those points without doing anything too complicated.

If I can't find a way to play this as it is by tweaking the config file, I guess that ioRTCW will do.

I haven't run the game widescreen in its vanilla state for a long time and unfortunately, I don't have access to a computer that can run RTCW in its original form at the moment, so I can't troubleshoot the issue for you. PCGamingWiki has a solution detailing what to do to play the game in a custom resolution unmodded:
https://www.pcgamingwiki.com/wiki/Return_to_Castle_Wolfenstein#Alternative_solution

There's also an entry detailing what to do when only a corner of the game appears.
https://www.pcgamingwiki.com/wiki/Return_to_Castle_Wolfenstein#Enlarged_upper_left_corner_of_screen

Again, keep in mind that the game will crash at many points during the campaign like this, but only when loading certain new levels IIRC. What you could do is quicksave right before you're about to exit a level. That way, if the game crashes on a new level load, you can change the resolution to something stable like 1024x768 temporarily, load up the quicksave, load into the new level in that resolution, and then go back and change it to your custom resolution. Annoying, but it works.

Hopefully the PCGamingWiki links fix your problem. If not, I'd just go ahead with the ioRTCW source port instead.

I tried both of those already but to no avail... eh.

Well, thank you!

Darn, well have fun anyway. I'll just post a couple other tips specific to the ioRTCW source port.

1) This is completely optional, but if you really want Steam to track your hours in the game, delete or move the existing WolfSP.exe file, and then rename ioWolfSP.exe to WolfSP.exe to replace it. If you do that, you will be able to start the ioRTCW source port through Steam like you normally would start the base game (through your Steam library), and Steam will track your time played. Steam looks for "WolfSP.exe" to run the game when you press that green play button on Steam, and by naming the sourceport file to that instead, it will just work and start up the sourceport file that's been renamed to it.
2) If you get any framerate issues, tick the graphics quality down to Normal.
3) If you notice the lipsyncing is off, open up the in-game console and enter the commands "/s_useopenal 0" and then "/snd_restart".

Oh I already knew the first tip because of Quake! hehe!

Thanks a lot, by the way, really! Have a nice day! :)

Open your config file and add...
seta r_mode "-1"
seta r_fullscreen "1"
seta r_customwidth "2560"
seta r_customheight "1600"
seta r_customaspect "2"

and then start the game

If anyone's interested, I solved it (and I think this fix should work for most people)...

I was looking in the Ghost Master's hub (which gave me the same problem) and I found you have to right click on the .exe, go to the compatibility section and then the high DPI settings button. Then tick the box below in the window you just opened (the one about the overriding), apply and you're good to go!! :D

https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=3271590967