Can You Hold the Stone that Wall of Stone Springs From

The example given in the spell specifies that "dirt" is ok. I feel as though this line is being taken out of the context it's written in. Here is the full context.

The wall doesn't need to be vertical, or rest on any firm foundation. It must, however, merge with and be solidly supported by existing stone. Thus, you can use this spell to bridge a chasm or create a ramp. If you create a span greater than 20 feet in length, you must halve the size of each panel to create supports. You can crudely shape the wall to create crenelations, battlements, and so on.

As long as the stone wall is merged with and coming out of a stone from which the wall can be made, it seems that are ok. I can't imagine that there exists a chasm which is pure stone without dirt, from which the spell is describing when it says you can create a ramp or bridge out of the wall.

As an evocation rather than a conjuration spell, it makes sense that the wall might require some stone to work with. This has been confirmed by Mike Mearls, that some sort of rock is required:

‏@wax_eagle : @mikemearls Can Wall of Stone be cast on dirt? It seems like its written to always need stone to merge to. Is that correct?

Mike Mearls : @wax_eagle that's correct - need some stone to merge with or be supported by. for instance, a bridge is OK if the two ends touch stone.

Daganev : @mikemearls @wax_eagle So no stone walls on a grassy knoll?

Mike Mearls :@Daganev @wax_eagle Correct, you'd need rocks of some sort (ruins, boulders) to support it.

It's up to the DM.

In the absence of a rule or feature calling for a saving throw, the DM decides if a circumstance calls for a saving throw. From the Dungeon Master's Guide, "Saving Throws":

Most of the time, a saving throw comes into play when an effect — such as a spell, monster ability, or trap — calls for it, telling you what kind of saving throw is involved and providing a DC for it.

Other times, a situation arises that clearly calls for a saving throw, especially when a character is subjected to a harmful effect that can’t be hedged out by armor or a shield. It’s up to you to decide which ability score is involved. The Saving Throws table offers suggestions.

The DM will decide if the particular situation merits a saving throw, and what the results of success and failure look like. For me, the creature's position on the bridge matters. Are they close to the end of the bridge? Probably saving throw. Are they in the middle of the bridge with nothing around to grab onto? Probably just fall damage. Your mileage may vary.