Can antacids cause ear crystals (which can cause dizzines)?

OctoniumOtoliths (also known as ear crystalsrocks) are a natural part of the vestibular system.

BPPV is caused by these otoliths coming loose, and so isn't directly related to calcium carbonate intake.

BPPV can result from a head injury or simply occur among those who are older. A specific cause is often not found. The underlying mechanism involves a small calcified otolith moving around loose in the inner ear.
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Within the labyrinth of the inner ear lie collections of calcium crystals known as otoconia or otoliths. In people with BPPV, the otoconia are dislodged from their usual position within the utricle, and migrate over time into one of the semicircular canals (the posterior canal is most commonly affected due to its anatomical position). When the head is reoriented relative to gravity, the gravity-dependent movement of the heavier otoconial debris (colloquially "ear rocks") within the affected semicircular canal causes abnormal (pathological) endolymph fluid displacement and a resultant sensation of vertigo. This more common condition is known as canalithiasis.

Further research into calcium supplements leads to a hypothesis that they cause milk-alkali syndrome, which has symptoms similar to BPPV:

The most common symptoms (of milk-alkali syndrome) are poor appetite, dizziness, headache, confusion, psychosis, and dry mouth; laboratory tests may show that a person with milk-alkali syndrome has high blood calcium, kidney failure, and metabolic alkalosis.

But the basic answer to your question: Can taking antacids (that have Calcium Carbonate) cause ear crystals to form and grow?, is no.

Calcium carbonate from antacids does not appear as calcium carbonate in the ear (or brain). When you take antacids, the gastric acid converts calcium carbonate to calcium chloride.

These effect is based on the chemical reaction CaCO3 + 2HCl → CaCl2 + H2O + CO2, in which hydrochloric acid and calcium carbonate, the base component, form water, carbon dioxide and calcium chloride, therefore neutralizing the acid by consuming the H+ radicals (JCED).

When absorbed, calcium chloride appears in the blood as calcium and chloride, the same way as sodium chloride appears as sodium and chloride.

Drugs.com does not mention any calcium carbonate side effects tat would be related to labyrinth or vertigo.

The search for "ear crystals" + "antacids" does not give any meaningful results.

If antacids were increasing the risk of ear crystals, this would be clearly mentioned in any serious article about ear crystals. It's not that crystals are formed because of too much calcium carbonate. Crystals are already there, they are part of normal anatomy, so everyone has them, but if they become loose, they can cause dizziness (Cleveland Clinic):

Three factors make it more likely that ear crystals may loosen:

• Age over 65 years
• Head injury
• Viral inner ear infections

Anyone who has vertigo that significantly affects his/her life can ask a primary doctor for advice. The doctor may recommend visiting an ENT specialist or neurologist. So, the first step is to get a proper diagnosis.

First of all: This is not an answer related to your described medical condition. Therefore, I will not draw any conclusions regarding your condition.

But, taking calcium carbonate containing antacids is very unlikley a cause for otoconia ("ear crystals") to form and/or grow. Actually, the formation of otoconia requires more than just a calcium carbonate source (1):

The fundamental requirements for the production of vertebrate otoconia are availability of calcium and carbonate ions. However, the activity of these ions in the endolymphatic space is far too low for spontaneous nucleation. Therefore, mechanisms for localized concentration of the reactants are required, a role typically played in other carbonate-based calcification systems by acidic glycoproteins.

Also, the formation of otoconia is a nonrecurring event in the human development (2):

Mammalian otoconia form at late embryonic stages [...].

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References:

(1) Development and maintenance of otoconia: biochemical considerations. https://www.ncbi.nlm.nih.gov/pubmed/11710459

(2) Mechanisms of Otoconia and Otolith Development https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4482761/