Pokemon Go can't be downloaded on Android 4.4

It could be that your phone really isn't compatible. Pokemon Go's system requirements are as follows:

Android

• Android 6 or above
• Preferred resolution of 720 x 1280 pixels (not optimized for tablet)
• Strong internet connection (Wi-Fi, 3G, or 4G)
• GPS and Location Services
• Rooted devices are not supported

iOS

• iPhone 6s or above
• iOS 13 or above
• Strong internet connection (Wi-Fi, 3G, or 4G)
• GPS and Location Services
• Jailbroken devices are not supported

_{Last updated March 2022, check Niantic's Supported Devices for up-to-date specifications}

With that being said, if this is your phone then its resolution of 480 x 800 pixels doesn't meet the minimum specifications

If Google says your device is not compatible, you will be unable to install Pokemon Go, even if you download the apk (the installation will fail with some sort of error).

However, there might still be a way for you to play Pokemon Go, but it is complicated and full of risks. In order to get it working, you would need to root your phone, install a newer version of Android on your phone (either the default one or a custom ROM like Cyanogenmod), install Google Apps (Gapps) on it if they aren't shipped, and finally install the Pokemon Go apk from an external source like Apkmirror or Aptoide, because Google won't let you download it even with a more up to date OS. But with the new OS you will be able to install it.

It works (I couldn't have run Pokemon Go on my device either because the system was too old and no updates were provided), but be aware that a rooted phone loses ots warranty and always do a full backup before doing/flashing anything risky!

If you can go this way also depends on if there are ROMs available for your device - if there is none, you can forget the idea.

Edit: You said in a comment you'd buy a new phone - if you would do that if it doesn't work anyways, its definetely worth a try, because rooted phones have many more advantages than just a new OS! You will also be able to highly cudtomize everything on your phone and using things like Xposed and XPrivacy (lets you restrict any unwanted permissions from any app) you'll have a great time! S3 smartphones should even have the newest official CyanogenMod ROM available... Go for it, there's nothing you could loose!