Your assumption is partly correct: depth of field is related to aperture, and smaller aperture increases depth of field.
With that said, the effect you associate with digital cameras is really the effect of focal length, which also affects depth of field: shorter focal length increases depth of field.
Digital cameras take "smaller negatives" based on the size of the CCD sensor array, and lenses are sized to be consistent with this. So, digital cameras typically have lenses with shorter focal lengths than, say, typical 35mm SLR.
This is somewhat an oversimplification, but you can rely on the relationships that increasing aperture or decreasing focal length increases depth of field.
As to "getting" depth of field on the G1, you will! You will have more than you might expect, if your experience has been with 35mm SLR. You might not be surprised if your experience has been with APS cameras and you were conscious of their depth of field.
In general, there is some zone that appears to be in focus because the extent of out-of-focus is too small for the eye to see. This zone is divided roughly one-third of it in front of the focus point and two-thirds behind it. What varies with aperture and focal length is how big (deep) this zone actually is.
Hope this helps.