Lots of confusion in this thread. A bipolar stepper motor REQUIRES two H bridges to drive it, but it also (unless it is a very low-power model with high winding resistance) also requires a current-control circuit. If you used an L298 as the H-bridge, you would typically use an L297 as the current chopper and step controller. I strongly recommend that you read the L297 datasheet in conjunction with the L298 datasheet (they are designed to work together), because of the understanding of (stepper) motor control it will give you.
The reason that a DC motor doesn't require a current-feedback chopper is that it has significant back-EMF while rotating. You don't need to control the current, because the mechanics of the motor kind of do that automatically: it will speed up until the back-EMF balances the supply voltage and the torque balances the current. A stepper cannot do that, so if you don't explicitly control the current then you will just overheat things.
If you buy a dedicated stepper controller like an A4988 or TB6560, the chip includes all three components: step-sequence control, current control and H bridges. Very convenient.
If you want to use the LMD18200 H-bridges to control your steppers, you can; you just need to provide an external circuit that does current control. You can do that very manually using a couple of comparators and a 555 or a flipflop or you could probably also make an L297 work for you here too. I haven't looked into the details so could be wrong about using the L297; I'm pretty sure it could be made to work though some external logic may be required to synthesize the LMD18200's DIR/PWM inputs from the L297's A/B/C/D outputs, and to make the current sense work properly.
You could also use an L6506 as the current controller, but you would still need to do your step sequencing manually, but that's not great chore. An L6506 just bundles up the comparators and flipflop that I mentioned in the previous paragraph.