I downloaded the Oracle SQLDeveloper, but when I opened it, it said that it requires a minimum of Java 8 and gave me the website for the download. I went on and downloaded Java
I had the same problem on Catalina 10.15.6 with SQLDeveloper 20.2.0 but none of the solutions mentioned here solved the problem. First I had to right click the SQLDeveloper.app file and choose "Open" as opposed to just double clicking to avoid the problem where it's not an Apple-approved app.
I tried installing various versions of the JDK and none worked until I tried JDK 11.0.8 (https://www.oracle.com/java/technologies/javase-jdk11-downloads.html). Looking at Applications/SQLDeveloper.app/Contents/MacOS/sqldeveloper.sh as mentioned by Robert Bain helped. It's clear from that script that this version of SQLDeveloper will only accept JDK version 1.8, 11, or 12.
Once I had JDK 11 installed, the install worked as expected.