No we're not. The data is clearly against this theory.

Coal was barely used until 1800s, early industrial revolution machinery used wood (indirectly solar power), charcoal, and river flow (indirectly solar power) instead. Oil didn't matter much until 1950s.

Amount of solar energy Earth receives annually is 3,850,000 EJ (and if we ever needed more there are ridiculously higher amounts of solar energy available is space). Human primary energy use is 487 EJ, or 0.01% of that. That's of course only because we conveniently don't count solar energy used to grow our food, and heat our planet - otherwise it would be fair to say human civilization uses 99.99% solar power (via photosynthesis, heating, water flow, wind etc.) and 0.01% all the other kinds of energy like fossil fuels, nuclear, geothermal etc.

We know fossil fuels were not necessary for industrial civilization because by the time we started using them we were already had industrial civilization. That's as good a proof as it gets.

History of ferrous metallurgy History of coal mining History of petroleum Solar energy

EDIT: Also, long before railways, river transport, and long distance sea transport were extremely common. If some place was inaccessible by water, canals were built. In fact rivers and canals, not railroads were the main mode of transport during the Industrial Revolution. Water transport requires ridiculously little energy compared to land transport, and requires no fossil fuels - even steam ships can use wood or charcoal. Oil wasn't even essential for transport - even as recently as during WW2 most armies relied on horses for carrying stuff around. Fossil fuels are completely irrelevant for explaining why industrial revolution happened. Transport during the Industrial revolution Horses in World War II