TL;DR: Google has ended the support for the Android Market app. Users need to migrate to the Play Store app by installing the compatible version (skip to the last section).
According to Android Authority, Google has ended Android Market support for Android 2.1 Eclair and below:
On June 30, devices running Android 2.1 Eclair or lower will no longer be able to access, or install other apps from, the Android Market.
In a different article, they also stated an opinion that the Android Market app would no longer be supported on "newer" versions:
Google also mentioned that it will continue to support “later versions of Android Market for as long as feasible.” Exactly what that means is anyone’s guess, but I think it’s safe to say that the company will eventually end support for Android Market on other versions of its operating system — and probably sooner rather than later.
The Play Store app is supported on Android 2.2 Froyo, and by default, the Android Market app should be automatically updated on supported devices when it's opened:
If you still have the Android Market on your device for whatever reason, it’s probably best if you upgrade it to the newer and much improved Google Play — provided you are running Froyo or newer. As soon as you open the Android Market app, a window will pop up saying that a new version is available for download. Then just follow the directions on your screen, and in a minute or two the Android Market will be replaced by Google Play.
However, according to some reports (1, 2), the Android Market shows an error message "A server error has occurred. Retry, or cancel and return to the previous screen." since at least 8 March 2018, preventing it to be updated to the Play Store app, and thus making it dysfunctional.
While the Android Market app cannot update itself, users are still able to install the Play Store app by downloading and installing the APK.
APKMirror listed all variants for the Play Store app, including for Android 2.2 Froyo. The latest version available on there is 4.9.13.
Version: 4.9.13 (80290013)
Min: Android 2.2 (Froyo, API 8)