About 8 weeks ago I stuck a couple cut fig branches in a glass of water in a sunny space. As of today, one of the cuttings has sprouted a couple of leaves out the side of the branch, but I don't see any roots developed at the bottom where initially cut. I would eventually like to plant the cutting outside. What do I do from this point on to make sure there is something I can plant that will be viable?
I think you need to do a few things. First, put the cut end into a watered 'airy' medium like vermiculite or pearlite or even sphagnum (orchid moss) -many other things meet the need but aren't so commonly available. It will also help to dip the cut end into rooting hormone. I know the big box stores carry RooTone powder that will do what you need. It is quite helpful, but isn't essential.
Then you need to make a humidity tent so the cutting doesn't desiccate. Moisture in the cutting will be drawn out through the leafs to dry air. The only way to slow this process is to keep the relative humidity of air around the leaf as close to 100% as practical. This is what a humidity tent does. Lastly, you want to keep it out of direct sun light.
There are many ways to gen up a little greenhouse/terrarium. I happen to have lots of 1 gallon nursery containers,so I partially fill (about half) with rooting medium, strike the cutting, and fit a one gallon plastic bag over the top. Some people often cut the bottom off a plastic soda bottle instead of a plastic bag.
Rooting seems to generally occur most rapidly at temperatures around 70F, so you could keep the struck cutting in its little terrarium inside your house on a north facing window sill if you want. I have a heater pad from an old water bed that I've rigged up in a shady spot outside to provide 'bottom heat'. I've gotten more success using that heater than without; again, it is quite helpful, but isn't essential.
Good luck (always needed with rooting cuttings).
Figs are very susceptible to fungal disease while rooting, so managing moisture well is important, and using a rooting hormone w a fungicide is a good.option. I typically cut the leaves to reduce moisture loss, but would prob not do that since you took them a while ago.
I greatly prefer taking winter hardwood cuttings of figs, as there is virtually no work required. If you girdle the branch a month prior to taking the cutting, it allows the auxins to build up and you don't even need any rooting hormone. Also you can root them directly in the spot they will grow, which is important as even after rooting there is significant loss during transplantation as the root systems develop slowly.
I know that is not a direct answer to your question but keep that in mind if your current venture goes south. Good luck!
A long time ago (before kids) i had visions of a fig cooperative, and wrote up some info on my experience w fig propagation in the DC area. You might find it helpful if you go w winter hardwood propagation.