The Greek of the English translations is the same as for the Latin Vulgate of Jarome, from which Douay Rheims is translated. There are a number of versions correcting the Latin influence (unbelief) following the RV "obeyeth not" for ἀπειθέω;
NRSV, ISV, NAB, CJB, WEB, DLNT: disobeys. NET, NIV "rejects". NJB "refuses to believe". As commentators like Cambridge commentary states,
he that believeth not] This may also mean he that obeyeth not, and this is better, for it is not the same word as ‘he that believeth’ with the negative added. The same correction seems to be needed, Act 14:2; Act 19:9; Rom 11:30 (see margin). Comp. Heb 4:6; Heb 4:11; 1Pe 4:17.
John 3:36
(THGNT) ὁ πιστεύων εἰς τὸν υἱὸν ἔχει ζωὴν αἰώνιον· ὁ δὲ ἀπειθῶν τῷ υἱῷ οὐκ ὄψεται ζωήν, ἀλλ᾽ ἡ ὀργὴ τοῦ θεοῦ μένει ἐπ᾽ αὐτόν.
(DRC) He that believeth in the Son hath life everlasting: but he that believeth not the Son shall not see life: but the wrath of God abideth on him.
(VulgCH) Qui credit in Filium, habet vitam æternam: qui autem incredulus est Filio, non videbit vitam, sed ira Dei manet super eum.
The word "apathy" derives etymologically from the Greek απάθεια (apatheia, "absence of feeling"), a term used by the Stoics to signify a condition of being totally free from the pathe, emotions and passions such as pain, fear, desire, and pleasure. The Greek verb ἀπειθέω (apeitheō) means "not persuaded" or "disobey" While the two words have similar roots in Greek, they have different meanings in English. Apathy refers to a lack of emotion or motivation, while ἀπειθέω refers to disobedience or being unpersuaded. See Apathy as philosophy. Beware of deriving doctrine from semantics or etymological connections.
Wiktionary :
ἀπᾰ́θεια • (apátheia)
want of sensation, impassibility
(of persons) insensibility, apathy
(among the Stoics) calmness, dispassionateness
Abbott Smith's Lexicon states: G544
ἀπειθέω, -ῶ
(< ἀπειθής),
[in LXX for H4784 מָרָה, H5637 סָרַר, etc.;]
as in cl. (MM, VGT, s.v.); to disobey, be disobedient: absol., Act 14:2; Act 19:9, Rom 10:21; Rom 11:31; Rom 15:31, Heb 3:18; Heb 11:31, 1Pe 3:20; c. dat., Joh 3:36, Rom 2:8; Rom 11:30, 1Pe 2:8; 1Pe 3:1; 1Pe 4:17 (Cremer, 475).†
Thus, the Latin Vulgate translation incredulus (unbelieving) is wrong in getting a contrast of believing vs unbelieving. There is a different Greek word for unbelieving G569 ἀπιστέω apisteo, which is not used. The contrast of believe vs disobedience/non-compliance, however, leads us to rethink the meaning of faith. It is not an agreement to a proposition, but obedience and compliance to a cause. Faith applies to the Son, as in obedience and submission, not a mental affirmation as traditionally believed.
It should be noted that due to the traditional dominance of Latin Vulgate, there is a great deal of the Vulgate's influence in the English translations, despite their base text was Greek. The famous example may be following Latin mistranslation of monogenes (only) as only begotten from unigenitus instead of unicus in John 3:16. The Latin translation also reveals Jerome's doctrine.