Can there be a standard way to handle this in all three platforms? Example would be using a single payment gateway for all. The app(on all platforms) asks for username, password, credit card info then submits to our server. Then our server authenticates and send to a payment gateway. When approved user is allowed to login to the App. Is that a good idea? Will it be allowed on all platforms?
Or does it require different approach for each platform?
On iPhone, this is not allowed. You may not avoid in any way apples standard mechanisms to pay for an app or to make in-app purchases.
There may be cases where an existing (paid) membership for a service might be complemented by a free iPhone app. For example, the free dropbox app will allow you to log in to your (paid) dropbox account. However the dropbox app does not rely on a paid account, it works with free memberships too.
As a rule of thumb: If you could handle payments through the app store, then you propably have to.