Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:
Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.
You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
If you've bypassed all the controls in the stove itself, and the voltage still drops to zero when you apply a load, then there's either a bad connection somewhere in the wire run, or one of your breakers is bad. (And breakers do go bad.) I'd replace the breakers, (fairly quick and cheap) and if still no luck then replace the wiring run from the panel to the stove.
Best Answer
You have more problems than just connecting up a dryer. Buy the way in answer to your question. NO you can't tap a range feeder and supply a dryer with it.
First you say your range feeder is protected by cartridge fuses. That would mean that you have and old Edison Base Fuse Panel or worse. In this state the AHJ recommend that when we as electrical contractors go into a dwelling it is our job to recommend the owner replace them.
Second NEC Article 220.55 requires a separate circuit for household ranges. The size is determined by the nameplate rating of the range. Since we don't know what size range the homeowner is going to install, we usually run a 50A circuit which covers all ranges up to 12KW. The minimum circuit it requires is a #8 wire and an over current protection of 40A. So you need to replace the 30A circuit you currently have.
Third NEC Article 220.54 requires a separate circuit to be run for the dryer. and It needs to be able to handle 5KW or the nameplate rating whichever is larger. So a minimum circuit would be a 30A circuit.
So if you have the capability in you panel to install a 50A circuit for your range. I suppose you could run a new range circuit and use the old range circuit for the dryer and relabel your panel to reflect the change.
Hope this helps and good luck.