BATTERY AMPS?

You're not doing to damage a VOM by reading voltage. That's what they are designed to do. You can however damage one when reading amps of current draw. That's why most VOMs have a 10 amp fuse.
I'm not sure what you are asking in reference to Ohms law. Ohms law is a formula that allows you to calculate any one of the 3 components of the law in a circuit if you know what the other 2 are.
*****Voltage = Current(amps) x Resistance(ohms)*******

To add to that, a voltmeter has a very high resistance so almost no current goes THROUGH it, just like no air goes THROUGH the pressure gauge on a compressed air tank. (Voltage is electrical pressure). The battery holds lots of potential energy but no current flows unless it has a path to flow through. That path always has some resistance just like standing on a garden hose makes resistance to water flow. An ammeter has very little resistance. When it is placed in a circuit, all current has to go through it, similar to a flow meter on that compressed air line. Current is limited to a safe value by the other resistance in the circuit such as light bulbs, wires, and motors. What you're asking basically involves part of a four-hour "class discussion" on about the third day of my 180-hour Automotive Electrical class. I can get anyone to understand electronics well enough to diagnose most problems, but there's a lot of preliminary stuff that takes the mystery out of the subject. By skipping over that you're still going to have questions gnawing in the back of your mind.