Cows have been pooping (and belching and farting) for millions of years, and they were not causing climate change then, and they are not, now. The cycle of life is a closed system, for all intents and purposes, and so no energy or material can be created or destroyed - so plants grow, and animals eat the plants, and the whole cycle stays stable over the long run.
It is factory farming that is the problem - we use natural gas to make artificial fertilizer - this is the source of the added carbon, that is contributing to warming the climate. The nitrogen in this fertilizer ends up as nitrous oxide - which is MUCH more powerful GHG than methane.
Cows (and other large herbivores) could actually save the planet, by sinking carbon into the soil.