My daughter asks if she can have some chips. My ex tells her "Only if you can run a mile on the treadmill in 8 minutes".
If your ex isn’t training your daughter to win a race, then this is a nasty thing to do to a child (if the story is accurate). It wouldn’t necessarily be inappropriate to ask them to run a mile, but the time limitation is the nasty bit. The message behind it (ie the reasons for the demand) isn't particularly good either.
My daughter naturally tries, fails, and then is angry that she is denied the chips.
Is she angry as not reaching the unrealistic goal your wife set, not getting the chips, or the intention behind your ex’s demand that she run a mile in 8 minutes, and the denying of the chips put together?
She yells and slams doors,
Is this inappropriate behaviour from a child
for what was just asked?
and my ex grounds her for 2 weeks from her friends and the computer.
For yelling and slamming a door? For getting angry at the parent making an unreasonable demand and the parent then punishing the child for not being able to meet the demand (by witholding the chips)?
She then calls me to tell me that I must continue the grounding at my house as well.
You want to support your ex doing the above? Remember that everything you do here sends two messages " one to your daughter, and one your ex. By supporting your ex in this : Your daughter learns that you support unfair systems, your ex learns that unfair systems are alright for her to impose.
In reality you wrote here because you know that you shouldn't be supporting her in this, but can't quite comprehend why, considering everything we are taught in the west goes against your feelings.
my relationship with my daughter is beautiful, mutually respectful, open, and appropriate
(I feel my exs' behavior is not appropriate)
Probably you are projecting your feelings to your child, but if it is accurate as described (and there’s no guarantee either will tell you exactly how it happened) you still need to teach your child how to handle such thing respectfully (both for the ex, and for herself)...they’ll be valuable lessons for her.
You will also need to explain to her why you are handling things differently to her mother " the reasons for it - be respectful to your ex when you do this, even if you don’t feel that way about her " she is still your childs mother, and how you talk about her will reflect in how your daughter deals with her as she grows up.
Just a note : usually it’s better for both parents to show a united front. This is fine where it’s just a matter of perspective (ie. Different ways of handling things with both being something the child can handle) " but it’s not fine when one parents actions are going to have a long term detrimental affect on the child, and they ask you to support that.
Another note : if you support your wife in this, you won't be able to teach your daughter how to deal with the matter respectfully...put yourself in your daughters shoes and think about why