Shorting the crystal
TLDR: The internal circuit should withstand shorts. However, transient static discharges by connecting "probes" can destroy nearly anything.
From the datasheet, the internal circuit is current-limited by a current source "I" fed from the DC supply Vdd:

When Vdd is 3.3V, DC voltage at XTAL and EXTAL pin should be near 0.95V. If so, chances are that the oscillator is still good to go. The oscillator merely stops oscillating while EXTAL is shorted to XTAL. The logic level may be undetermined at about 0.95V, which may confuse circuits connected to EXTAL downstream.

Shorting XTAL to EXTAL should neither damage the MOSfet, the current source, or the external crystal. The current source "I" seems large @ 2mA spec in the data sheet.


Shorting either XTAL or EXTAL to ground is current-limited to 2mA from the oscillator's current source "I"....no problem.
It is unlikely that you've connected EXTAL to Vdd, but that's no problem...again, current is limited by the current source "I". When  EXTAL goes to logic high, XTAL should go from 0.95V to logic low. Doing this test would satisfy me that the oscillator is working OK.