If we assume that a single particle has penetrated the skin on your finger, possibly beneath the epidermis layer (meaning it can't be shed through the normal skin shedding process), are there any other ways the body can naturally remove it? Glass is inert...
The vast majority of glass is inert, yes, but it's not sterile, and, as such, will lead to an inflammatory reaction encapsulating and eventually (if it's not deeply sub-dermal) expelling it with the outward growing epidermis and shedding keratinocytes.
Could it enter a blood vessel and migrate if it's very small?
Not likely. A shard of glass is very good at cutting/penetrating, but not very good at traveling due to its shape; it gets snagged on things. A very small shard of glass might lacerate or puncture a capillary or venule, but its shape and size would prevent most tiny shards of glass from entering the bloodstream and traveling remotely. Keep in mind that the internal diameter of an epidermal capillary is only 4–6 μm, and that of a collecting venule is 10–15 μm. While it's possible to have a shard of glass of 10μm, it would likely just injure the vessel, causing an inflammatory reaction that would wall it off and probably obstruct/close off the vessel. The only way I could think of to have glass shards in circulation would be to inject them directly into a vein, which has a comparatively large lumen.
if it really moves in the vessels, is there a natural way how the body could naturally eject it with time?
Assuming it is injected directly into a vein, it will get stuck somewhere in the circulatory system and will, through injury, cause an inflammatory response. It will get walled off, and walled off objects not directly under the skin or some surface (alveoli, gut) will just stay put, or the inflammation might cause a problem that the body deals with in another fashion.
...but how can the compounds constituting blood in vessels dissolve SiO2??
It doesn't get dissolved in blood; blood is relatively neutral. It gets dissolved over a long time in the inflammatory-induced encapsulation. How, exactly, I can't say, but I highly doubt that the pH in an environment where free radicals are produced is neutral.
I want to add that the danger from glass shards injected IV is mostly hypothetical, and has not to my knowledge been studied in vivo except once in 1947. The abstract in that paper states
This report on the effects of glass particles when injected into animals indicates that massive doses are required to produce damage to the organs examined during the study.
This is no reason not to use caution when aspirating fluids from glass ampules. Though I have never seen a problem from injecting such fluids locally (I wouldn't expect to be able to see what happens when injected into an IV as it would be remote), why not err on the side of caution? (Fyi: Rubber particles from needle insertion into rubber-stopped ampules results in loads more rubber microparticles in the fluid than glass in a single-use ampule.)
The role of macrophages in the resolution of inflammation
An In Vitro and In Vivo Study of Glass Particles in Ampules