I'd say: "he deliberately kept me waiting" -- to keep someone waiting [on purpose/deliberately].
Also possible but not quite as safe I would say (only possible in certain contexts): "he had me wait" -- to have someone wait

Ghol ‹GB›
to keep so. waiting hab ich ebenfalls, direkt aus LEO.

Für "... had me waiting" gibt's aber gleichermaßen viele Einträge wie für "... have/had me wait" - bei Google;

Ist die erste Variante dann genauso unsicher wie die zweite und man bleibt besser gleich bei "he kept me waiting"? Weil ich halt bei LEO so gar nix dazu gefunden hab ...

Jutta - It's hard for us native speakers to come up with rules.
But here are a few examples:
He kept me waiting at the cafe for hours while he organized his CD collection both alphabetically and chronologically.
She kept him waiting downstairs while she put the final touches on her outfit for the evening.
He has me wait for him at the train station since that's directly on his route home.

I'd say: both versions are possible:

to have s.o. do s.th. AND
to have s.o. doing s.th. ,

and both practically mean the same, i.e. to order or arrange (that s.o. does s.th) or to cause s.o. to do s.th.:

I'll have the gardener plant some trees.
Have the driver bring the car round at 4.

She had her audience listening attentively.
The film had us all sitting on the edges of our seats with excitement.

Reinhard: For me your examples are totally different grammatical constructions. "to have someone doing something" is a totally different sense from "to have someone do sth.", so I find it very misleading to say 'both versions are possible'. Your examples are correct though.
How would you translate the "had us sitting attentively" or "had us sitting on the edge of our seats" idea into German? The construction "jmdn etwas tun lassen" surely isn't right.

Jutta: The phrase "to have someone wait" exists. As I said, I would recommend "keep someone waiting" as the safest option, if you mean a situaiton where someone arrives late at an agreed meeting place [either deliberately showing disrespect, being impolite, or unintentionally], thus forcing the other person to wait.
An example of the two phrases: you have an appointment with your boss at 10.00. You are sitting outside his office, he comes out at 10.05 and says: "sorry to keep you waiting, an important customer just phoned." (Not "have you waiting").
Then you are talking with him in his office, and his wife phones him with a private issue he has to discuss, he might then "have you wait outside" (ask you to wait outside) for the duration of the phonecall (not "keep you waiting outside").

Ghol: Bitte keine Unterstellungen! Ich habe nicht behauptet, dass beide Versionen gleich sind, sondern dass beide praktisch die gleiche Bedeutung haben. Eine deutsche Übersetzung habe ich überhaupt nicht erwähnt. Ich habe darauf hingewiesen, dass beide Versionen im Englischen möglich sind, und die Beispielsätze belegen das.
"Jemanden etw. tun lassen" ist _ Deine_ Übersetzung. Ich würde hier immer mit "veranlassen" übersetzen. Verstehst Du, was ich meine?