If fossil fuels are that depleted, following choices can present themselves:
We can go back to fighting like the way the Romans did. Good old mano to mano hack and slash. Civilization would collapse anyway to medieval style.
But before that can happen you would need to exhaust every drop of oil, along with other means to generate it. There are still reserves of oil. What is being depleted is oil that can be cheaply obtained because there are existing wells or its easy to drill. There are more oil reserves out there, but because of where they are located, it would cost more money to obtain each drop. There are oil reserves out there that won't be viable for exploitation, for example, until oil reaches $200 a barrel, and even more if the prices reach $300 and $400 a barrel. When a certain price point is reached, it would be more economically viable to synthesize fuel out of coal, limestone, sandstone and shale rock. This is actually good for the US, Germany and China, the three having the largest deposits and production of coal. Nazi Germany was trying to synthesize fuel out of coal during the big war.
The result won't be that fossil fuel powered platforms still won't disappear, but it becomes more and more expensive to operate them. The greatest impact would be on the gas guzzlers, platforms that require gas turbines to operate them.
For example with tanks, gas turbine powered tanks like the M1 series would be seriously affected, but it won't be that bad to the relatively more efficient diesel powered tanks the rest of the world uses. The US Army may have to reconsider to re-engine their tanks to diesels or build the next version of Abrams with diesel engines. Diesel fuel is relatively easier to synthesize out of coal, and there are other means to produce it, like bio-fuels.
Ships will have to go out of their CODAG propulsion, into nuclear-steam power, coal fired fired steam power, or pure diesel (CODAD). Submarines can go either nuclear, fuel cell AIP, or Stirling. Even conventional SSKs with diesels have been relatively efficient at the start. Submarines would probably be the least affected of all naval combatants.
Air power would be the most seriously affected because it has no other choice. There is also the double whammy of vastly increasing platform cost with each generation of manned fighters that they may price themselves out of the picture, replaced by stealthy UAVs and cruise missiles. Unless some new technology replaced fossil fueled gas turbines, conventional manned air power may obsolete itself out of existance, or go back to gliders and propeller aircraft. With that gone, so are air bases and aircraft carriers as well, and the carrier battle group as we know it.
But rocketry on the other hand, won't be scratched, but kerosene powered boosters will be phased out. Cruise missiles will still be there but powered with some other liquid fuels. Wars will be decided by surface launched missile power in conjunction with UAV support. We will skip air power for space power, since we can launch satellites without the use of fossil fuel, though we need to eliminate kerosene powered boosters. The combination of ballistic+cruise missiles cued by satellites will replace the conventional bombers and strike fighters, and in response, interceptors will be made against such satellites, forcing the first wars in space.
Based on this, you can create a scenario of late 21st century war. I can imagine navies for example, using large nuclear powered battleships and arsenal ships packed with VLS launchers for long range cruise missiles and ballistic missiles. Some of these may be intended to strike at surface targets, others at sea targets, but can also be directed against targets in space. They can probably launch their own satellites too. Destroyers, frigates, corvettes and FACs will still all exist, though gas turbines will be replaced with other engines, and more so the nemesis of all surface ships, the submarines, the most unchanged species of all.
Its possible to draw a scenario where modern technological warfare is still possible, albeit by combining the twin strategies of making fuel usage much more efficient and thrifty, along with substitution. using nuclear power, coal, synthetic, biofuels, and non-fossil fuels.