Why is it that Americans invite all and sundry to "Enjoy!" without including an object noun in the sentence, when the dative case of the verb 'to enjoy' clearly demands one to be correct?
Enjoy is used as an absolute verb here, which means that the object is implied but not stated.
There is another famous example also: if looks could kill... Here, the transitive kill is used as an absolute verb.
Though, there is a history behind the good wish "Enjoy!" meaning "Enjoy your meal!" and it is derived from a Yiddish phrase. An article in The New York Times talks about the history of this phrase and mentions the first absolute use of the transitive enjoy:
In his 1986 book, "Yiddish and English," Random House lexicographer Sol Steinmetz cites this 1968 quotation of furrier Jacques Kaplan by Marylin Bender in The New York Times: "It's a dancing over the volcano attitude, an enjoy-enjoy philosophy." That reduplication is typical of Yiddish -- Es, es means "Eat, eat" -- and the friendly command of Enjoy! comes from Hob anoe , a Yiddish phrase derived from the German hob , "have," and the Hebrew hanoe , "enjoyment."
The earliest example of the absolute use of the transitive enjoy comes from an essay by English author John Ruskin in "The Eagle's Nest," in 1872: "It is appointed for all men to enjoy, but for few to achieve."