Is TCP packet filtering possible on mobile platforms?

iOS

I don't think it is possible on iOS.

I didn't find a public API for network monitoring/packet filtering. There is a possibility that such API exists but it's hidden. But in that case Apple App Store review guidelines states:

2.5 Apps that use non-public APIs will be rejected

If you need one specific quote to show that it is not possible, you can use this:

iOS does not support packet tracing directly. However, if you connect your iOS device to a Mac via USB...

from official Apple Technical Q&A QA1176.

Alternatives

The next best thing is to a configure a proxy server manually in Settings and then filter the traffic on the server-side. Running the proxy locally, on the device is not an option because of limitations of iOS background tasks:

2.16 Multitasking Apps may only use background services for their intended purposes: VoIP, audio playback, location, task completion, local notifications, etc.

Also, this post suggests it might be possible to set-up a VPN connection programmatically on iOS 8. It would also require to send the traffic of the device and I'm not sure about compliance of this method with guidelines.

Non-alternatives

Some apps provide functionality of measuring the network traffic. But they use dedicated API for network statistics: iPhone Data Usage Tracking/Monitoring.

There are also ways to packet trace on iOS via USB cable described here.

Android

On Android you can configure the device to use your app as a VPN service. But:

• It requires you to display a dialog describing the consequences of giving a permission to act as a VPN.
• You have to show a persistent notification while VPN is active. An example of app that does it is tPacketCapture.

To ask for user permission, you call VpnService.prepare:

public void onClick(View v) {
     Intent intent = VpnService.prepare(getApplicationContext());
     if (intent != null) {
          startActivityForResult(intent, 0);
     } else {
          onActivityResult(0, RESULT_OK, null);
     }
}

and handle the result, starting your VpnService.

protected void onActivityResult(int requestCode, int resultCode, Intent data) {
    if (resultCode == RESULT_OK) {
        Intent intent = new Intent(this, MyVpnService.class);
        startService(intent);
    }
}

Your VpnService have to implement public int onStartCommand(). The service is treated as a foreground service and should not get killed by the OS.

This question: Android VpnService to capture packets won't capture packets and it's comments shed some light on the packet handling itself.