Is it possible to write English in Greek script? Would it look better?

This could go one of two ways, neither pretty.

You could phonetically transcribe English into Greek, Ancient or Modern, using the phonetics of the Greek alphabet unchanged. As Konstantinos Konstantinides says, that would sound horrible, because it really would be English with Greek vowels and consonants.

In fact, when Greeklish ( Greek in ASCII) was a going concern online, a popular party trick was to drop in some English, transliterated into Greek, but in Roman characters. That should give you a flavor of the ugliness.

Ιφ γιου φάιντ δις βέρι χάρντ του ριντ, δεν γιου γουΐλλ αντερστάντ δατ περχάψ τρανζλίτερεϊτεντ Ίγκλις ιν Γκρικ κάρακτερζ ιζ νοτ α λάικλι άουτκαμ.

If giou fai”nt dis beri xarnt tou rint, den giou gouill anterstant dat perxay tranzliterei”tent Igklis in Gkrik karakterz iz not a lai”kli aoutkam.

Looks a bit like Tok Pisin, only with velar fricatives. An Attic transliteration would not fare much better.

Ἰφ ἰοὺ φαίνδ δὶς οὐέρι ἃρδ τοὺ ρίδ, δὲν ἰοὺ οὐὶλλ ἀνδερστάνδ δὰτ περὰψ τρανσλίτερητεδ Ἴγκλις ἰν Γρὶκ κήρακτερς ἰς νὸτ ἀ λαίκλι αὔτκαμ.

The other alternative would be to use the Roman alphabet as a transcription, one to one, as José A. Ugalde σuggests. There is precedent for this; in fact, the Greeklish I use does this (which is why I had <y> for psi and <d> for delta above). But it would be even sillier.

Some of you will have seen this before too, in the 90s: it would be merely English text typed in Symbol font.

Ιφ υοθ φινδ τηισ ωερι ηαρδ το ρεαδ, τηεν υοθ ςιλλ θνδερστανδ τηατ περηαπσ τρανσλιτερατεδ Ενγλιση ιν Γρεεκ ψηαραψτερσ ισ νοτ α λικελυ οθτψομε.