Nowadays, among it's meanings it has this one:

hail verb

1. [trans.] call out to (someone) to attract attention: the crew hailed a fishing boat.
2. signal (an approaching taxicab) to stop: she raised her hand to hail a cab.

The Archaic usage is signalled as follows:

• exclamation archaic
  expressing greeting or acclaim: "hail, Caesar!"

The Etymology is this one:

ORIGIN Middle English: from the obsolete adjective hail [healthy] (occurring in greetings and toasts, such as wæs hæil: see wassail ), from Old Norse heill.

which is related (it surprised me a bit) with Whole and its etymology:

ORIGIN Old English hāl, of Germanic origin; related to Dutch heel and German heil, also to hail (the quote I pasted above this one). The spelling with wh- (reflecting a dialect pronunciation with w-) first appeared in the 15th cent.

Nowadays, among it's meanings it has this one:

hail verb

1. [trans.] call out to (someone) to attract attention: the crew hailed a fishing boat.
2. signal (an approaching taxicab) to stop: she raised her hand to hail a cab.

The Archaic usage is signalled as follows:

• exclamation archaic
  expressing greeting or acclaim: "hail, Caesar!"

Considering this, we can say that "Hail to the King!" can be both a way to express acclaim, praise to the King and express greeting.

But we even might be able to say that it can be both together. Deciding which one of these is the correct meaning depends on the context, as it usually happens with words with different acceptions.

The Etymology is this one:

ORIGIN Middle English: from the obsolete adjective hail [healthy] (occurring in greetings and toasts, such as wæs hæil: see wassail ), from Old Norse heill.

which is related (it surprised me a bit) with Whole and its etymology:

ORIGIN Old English hāl, of Germanic origin; related to Dutch heel and German heil, also to hail (the quote I pasted above this one). The spelling with wh- (reflecting a dialect pronunciation with w-) first appeared in the 15th cent.