Can an Astral Self monk using their Word of the Spirit ability cause the Verbal component of any spells they may be able to cast to be audible only to themselves?
Yes, but this doesn't remove the need to speak the words.
Sure, you can cast a spell using the Word of the Spirit ability to make the Verbal components of their spell only audible to themselves (or to one other creature within 60 feet of themselves). However, you're still required to speak the words of the Verbal component - your mouth still moves. Additionally, you still need to perform any Somatic components to the spell, as well as have any Material components at hand. If anything stops you from performing any of the components (e.g. including having your mouth gagged), you're unable to cast the spell.
This would likely make it apparent that you're casting a spell to anyone who's looking at you, especially when the spell promptly takes effect immediately afterwards. However, it would allow you to cast a spell with a Verbal component while hiding without immediately giving away your position (though the effects of the spell might).
Given that to acquire this ability you'd need six levels of Monk, and then acquire the ability to cast spells somehow (likely through multiclassing or racial/feat abilities), it's unlikely that it'll break anything. You'd be casting 1st-level spells silently, while your party members would be casting 3rd- or 4th-level spells normally.
Maybe? I'll quote the relevant two bits of rules:
Word of the Spirit. When you speak, you can direct your words to a creature of your choice that you can see within 60 feet of you, making it so only that creature can hear you. Alternatively, you can amplify your voice so that all creatures within 600 feet can hear you.
What we can infer from that is that the words are still SPOKEN, since the target still HEARS them (and not receives them via mind etc). This means that by MY interpretation as a DM, RAW would allow this exploit since this is the text for somatic spellcasting:
Most spells require the chanting of mystic words. The words themselves aren't the source of the spell's power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can't cast a spell with a verbal component.
However, this doesn't mean that you're secretly casting the spell. That's something that sorcerers have to spend metamagic points for, and is not included in Word of the spirit's description, so it doesn't do that. People will still notice you're casting a spell. I'd rule they have disadvantage on the arcana check knowing what spell it is etc, but that's about it.