I used to do R&D for one of the companies who make a few of the well known brands of washing powder so I'll add my two penneth.
Its unlikely you did any damge to your car's clear coat finish. (Its the clear coat which gives your car its nice shiny finish.) Here's why.
The only things in there which *might* damage your finish are bleaches and zeolite builders. The bleach which is used is pretty weak and would need a fair amount of time to actually attack the finish. It might be a problem if you soaked your car in it for a few hours at 60°C but a couple of minutes at ambient won't do it any harm.
Zeolite builders are the only non soluble solids found in a washing powder, typically they are 1-5 microns in size - pretty tiny- you're more likely to scratch the finish by removing the crud which has attached to your car than with the zeolite found in the powder.
The surfactants in the washing powder are responsible for its cleaning ability and I pretty much guarantee that you'll find the same surfactants in a car shampoo too, for those interested they are alkyl benzene sulphonates and an ethylene oxide based non-ionic surfactant probably C12- 7EO plus a bit of perfume to make you think that you're getting something special.
There isn't acually any 'salt' in your washing powder. It is stuffed full of sodium sulphate but that won't harm you finish. (and for all you smart@rse chemists out there by 'salt' I was refering to sodium chloride

) There's also a hell of a lot more salt kicking about in winter. I'd be more worried about that.
Bear in mind that the companies who develop these products put them through any number of tests to make sure that they will hold up to pretty much anything. Car finishes are some of the most highly developed coatings around, serious amounts of money have gone into developing them, they'll have done this test.
The only thing you've done (apart from removing the crud) is to remove a little bit of wax from the surface (assuming you've waxed the car). All you've got to do (other than to use a good quality car shampoo next time e.g. Autoglym) is to give it a wax with a good quality wax e.g. Autoglym red label, then Autoglym Gold label.
Why use an actual car shampoo next time? The washing powder probably isn't powerful enough to remove some of the road film which has built up on the car. This comes from unburnt fuel, engine oil etc.
The snippet which Stormcrow copied from another site isn't actually correct (no offence meant btw) and if this is from a car care website then they should be ashamed of themselves for providing very poor advice.
The reasons its shiny is not because of 'oils' on the surface, its because its got a very smooth surface and can reflect a large proportion of the incoming light. All the wax does is to restore that smooth finish by filling over the microscratches on the surface. The smoother the top layer, the better the gloss.
Carnauba waxes such as those in Autoglym Gold and Meguiars and products such as Zaino (which aren't technically waxes) can be applied into those microscratches and its the ability of these products when buffed to form a coherant film which gives then gives the car its excellent highg gloss finish.
Washing-up liquid/ powder won't/ can't cause oxidation, this is a chemical and process and there's nothing which can cause an oxidisation reaction other than the bleach which I don't believe would.
How do I know about car finishes I hear you ask? I now do R&D for the surface coatings industry