The /ʒ/ sound is spelled with <g> or <j>. The /h/ sound is spelled with <h>.
This is a contrast between a voiced alveolar fricative and a voiceless glottal fricative. Since /h/ can only occur initially and /ʒ/ only medially or finally except in words of French origin such as gendarme, no pairs showed up in the dictionary search. I have identified several that might occur, though it could be disputed whether these are genuinely English words.
gîte heat gîtes heats Jacques hack jolie holly