Using hydrogen instead of hydrocarbons would certainly eliminate the CO2 production but burning the fuel mostly used today, JET A-1, already produces about 1.4 tonnes of water vapour per tonne of fuel burned. Atmospheric O2 contributes the additional mass. The process also creates over 3 tonnes of CO2. Variations in existing atmospheric water vapour content determine whether contrails form and indeed whether they persist. Interestingly if the right conditions exist for contrails to persist, vast amounts of additional atmospheric water vapour will condense and freeze on the ice crystals that make up contrails. Some measurements have determined that 1 tonne of burnt jet fuel can lead to 200 tonnes of contrail forming.
I'm still researching this next statement but...my understanding of contrail formation is that the formation process is dependent on the presence of particles of soot in the exhaust plume that act as condensation nuclei. No hydrocarbons, no soot, no contrail.