As others have mentioned, the header goes on top of your jack studs, not between them. The king studs are the full length studs that are either added to get the proper width opening, or your existing 16" OC studs. When removing studs, before the header is added, you'll need to reinforce the area with an adjacent load bearing temporary wall to prevent sagging or a possible collapse depending on what's above.
The sizing of the header is important and will depend on the load. You may only need a pair of 2x6's installed on edge with a plywood filler, while a pair of 2x8's will likely be more than enough, but you'll want to seek professional advice on this.
I think the bigger question is if you actually want to do this. You'll need to find somewhere to run the water line, install an outlet, and you need to be sure that the fridge is properly vented (they aren't creating cold, they are moving heat, and need to be able to exhaust it into your home). If anyone pushes the fridge too far back, it will likely go right through the unsupported drywall behind it. And if this is an exterior wall, you'll be compromising the insulation and vapor barrier.