A new version of the legacy launcher has been released that solves this issue. If your launcher does not automatically update, you can download the updated version (1.6.93) here.

If you're able to launch the updated launcher (2.x.x), it is recommended to update to this one instead. You can find the download links for it here. If you're not able to launch the 2.x.x launcher, please create a new ticket about that.

You find the workaround for this issue that could be used before it was fixed below. It is recommended to use the updated launcher (legacy or not) instead of using the workaround below.

This is an error that occurs only when launching 1.14.4, 19w36a, or versions that have been released after 19w36a. It happens because the legacy launcher does seem to have problems with downloading and installing the newer version files.

To work around this issue, you can install versions you want to play manually. This is a rather tedious task. If you can use the updated launcher, or a third party launcher (such as MultiMC), you might want to use those. (Beware of pirated third-party launchers though, they might steal your login data.)

How to install a version manually

1. Close the launcher.
2. Visit the version page of the version you want to play on the official Minecraft Wiki.
3. On the top right of the page, you'll see a box with some informations about the version. Here, you find links to download the version files directly from Mojang's servers. Click the "Download → Client" and "Download → .json" links to download a client.jar file, and a <version>.json file.
4. Navigate to your .minecraft folder.
5. Navigate into the folder versions.
6. Create a new folder. You may choose a custom name, but not the name of any vanilla version. For instance, I would recommend to use 1.14.4-fix if you want to install 1.14.4.
7. Copy the client.jar file into the folder you just created and rename it so that it has the same name as the folder. For instance, for 1.14.4, that would be 1.14.4-fix.jar.
8. Copy the <version>.json file you downloaded in that folder as well, and rename it so that it has the same name. For instance, for 1.14.4, that would be 1.14.4-fix.json.
9. Open the .json file in a text editor of your choice.
10. Edit the "id": "<version>" field so that it now has the same name as you gave the version, for instance "id": "1.14.4-fix".
11. Delete the first downloads object ("downloads", and everything inside of the curly braces behind).
12. Save the file and close it.
13. Open the launcher.
14. Edit your launcher profile and select the version you just installed, for instance 1.14.4-fix.

Happy mining and crafting!

If you have trouble following these steps above, please do not comment on this ticket. Instead, I've opened a thread on Reddit if you need support.

I'm aware that this workaround is very tedious and hard to follow, but it's the only one that is found to work on computers that can't use the new launcher as of now.