This is a more complicated question than the binary alternative posed by OP's title question would suggest. In fact, the KJV here does a good job of reflecting an ambiguity in the Hebrew that almost all other translations obscure (including the NJKV).
Here is the Hebrew text of 2 Kings 8:11 with a literal gloss:
and he stood his face
and he set (?) until shame
וַיֵּ֖בְךְּ אִ֥ישׁ הָאֱלֹהִֽים׃
wayyēbk ʾîš hāʾĕlōhîm
and the man of God wept
It should be clear from this literal rendering of the Hebrew into "English" that there is no verb of seeing here. Nor does the verb in the middle line have an object, which is odd for a verb meaning "set, put" ("set or put what?", we need to ask here). And further, the sense intended by ʿad-boš ("until shame") also remains opaque (whose "shame"? "ashamed" why?? is this idiomatic???).
The hiphil form of ʿmd with the object "face" is unique to this verse in the Hebrew Bible, and as Cogan and Tadmor note (in II Kings: A New Translation... [Anchor Bible 11; 1988], p. 90) this verb (in this form)
together with pānîm, can only mean "to make the face stop, keep motionless, expressionless."
This understanding is reflected in the translation of the JPS Tanakh which reads here:1
The man of God ckept his face expressionlessc for a long time; and then he wept.
And it adds this footnote: "c-c Meaning of Heb. uncertain."
Nearly universally, however, the notion of seeing or staring is added to make some sense of this otherwise opaque vignette.
To sum up to this point: it is not clear exactly what action is being undertaken ... and we haven't even got to OP's main interest:
Who is doing what?
Gwilym Jones (1 and 2 Kings, NCB [Eerdmans, 1984], vol. 2, p. 444) sets out nicely the main options with their rationale for who might be the subject of the first two verbs in the verse (the last one, "wept", has an explicit subject, of course):
Hazael is the subject of both verbs. Why? Because v. 10 (the prophet's announcement) expects a response from the one who sought the prophet's guidance = Hazael.
Elisha is the subject not only of the last verb, but of all three. Why? Verse 11 continues on from v. 10 with no change of subject, and Elisha is the subject of the verb ("And Elisha said to him...") in v. 10.
"To assume a change of subject within the verse, as is clearly brought out in the NIV, 'He stared [i.e., Elisha] at him ... until Hazael felt ashamed'." Why? (Jones doesn't offer the same kind of rationale here, but...) The last clause offers an explicit subject, and this implies a contrast with the preceding which must then be Hazael — but the first verb carries the same subject as the preceding verse, and that's Elisha.
Ultimately, and unfortunately, there is no "knock-down argument" here. Not only is the action obscure, but its actors are ambiguous. Translators and commentators across the centuries — since at least the time of Josephus (Antiquities, Bk. IX.90) — have filled in these narrative gaps as best makes sense to them.
To me, the most appealing reading in light of the state of the Hebrew, is to unpack the ambiguities in this way:
- in v. 10, Elisha delivers his message (also having ambiguities in Hebrew), and then in v. 11 keeps a "poker face",
- while Hazael begins to squirm, in contrast to Elisha who...
- ...who gives in and weeps, now identified as "the man of God" (= Elisha),2 as he comes to understand what his message means for Israel (v. 12).
- So too Marvin Sweeney, I & II Kings: A Commentary (Westminster/John Knox Press, 2007), p. 314.
- This may well have something to do with the patterns identified by E.J. Revell, The Designation of the Individual: Expressive Usage in Biblical Narrative (Kok Pharos, 1996). See the discussion in ch. 12, "The Designation of Priests and Prophets".