calcium deposits in a kettle and calcium kidney-stone crystals in the body.

 There are two different things here: calcium deposits in a kettle and calcium kidney-stone crystals in the body.

For boiling-water calcium sediment, yes—vinegar can help. 

Apple cider vinegar contains acetic acid.

 Hard-water scale is largely calcium carbonate (CaCO₃). 

Acetic acid reacts with calcium carbonate, converting it into more soluble calcium salts while releasing carbon dioxide:

CaCO₃ + 2 CH₃COOH → Ca(CH₃COO)₂ + H₂O + CO₂↑

That's why vinegar can dissolve the white calcium scale in a kettle.

But inside the human body, it doesn't work the same way. 

Drinking apple cider vinegar does not directly travel to the kidneys and dissolve calcium crystals like vinegar dissolves scale in a kettle. Kidney stones are commonly calcium oxalate or calcium phosphate, and the body's blood and urine chemistry tightly regulates acidity.

There has been some research suggesting dietary vinegar/acetic acid may be associated with changes in urinary citrate and calcium metabolism, but evidence that apple cider vinegar actually prevents or dissolves kidney stones in humans is limited. 

Lemon/citrate has considerably better clinical evidence for increasing urinary citrate in appropriate patients.

So, in simple terms: vinegar + kettle calcium scale = can dissolve it; drinking vinegar + calcium kidney stone = not the same chemical process and shouldn't be expected to dissolve the stone.

human clinical studies that support the statement, but there is an important update: vinegar has some promising evidence, so saying there is no evidence for vinegar would be too strong. The evidence for citrate/potassium citrate is substantially stronger, and lemon juice has multiple human studies.

Strongest evidence: potassium citrate

A randomized, double-blind, placebo-controlled trial followed patients with recurrent calcium stones and low urinary citrate. Patients taking 30–60 mEq/day potassium citrate had their stone-formation rate fall from 1.2 to 0.1 stones per patient-year. About 72% went into remission, compared with 20% receiving placebo. Potassium citrate also significantly increased urinary citrate and urine pH.

Randomized double-blind study of potassium citrate — PubMed

Lemon juice / lemonade evidence

A 1996 clinical study found that lemonade therapy more than doubled urinary citrate in people with hypocitraturic calcium kidney stones. The authors concluded that lemonade could be a useful adjunctive treatment.

Dietary manipulation with lemonade to treat hypocitraturic calcium nephrolithiasis — PubMed

A later study of recurrent calcium oxalate stone formers also found that lemonade therapy increased urinary citrate and urine volume, both favorable factors for stone prevention.

Lemonade therapy increases urinary citrate and urine volume — PubMed

There is also a larger randomized trial of fresh lemon juice supplementation in recurrent calcium oxalate stone patients. The investigators found evidence suggesting lemon juice added to a standard diet might reduce recurrence, although adherence declined over time, making the long-term result less certain.

Fresh lemon juice supplementation randomized trial — PubMed

However, lemon juice isn't equivalent to medication. A crossover trial involving 21 stone-forming patients found potassium citrate significantly improved urinary citrate and pH, whereas lemonade did not significantly do so in that particular study. Lemonade did help patients maintain greater urine output.

What about apple cider vinegar?

There actually is some supportive research for vinegar, but it's much less established.

A 2019 study reported that a pilot clinical component found daily dietary vinegar was associated with higher urinary citrate, lower urinary calcium, and reduced calcium oxalate stone recurrence. The researchers also performed mechanistic experiments investigating acetic acid's effects.

Dietary vinegar prevents kidney stone recurrence — PubMed

But this does not establish that drinking apple cider vinegar dissolves existing calcium stones the way vinegar dissolves calcium carbonate scale in a kettle. More recent literature still describes the risks and benefits of apple cider vinegar for kidney-stone management as uncertain and calls for additional research.

Interestingly, there is now also a specifically designed study titled “Apple cider vinegar for prevention of urinary lithiasis (APUL)” evaluating ACV's effects on 24-hour urinary citrate, pH and volume.

Bottom line

The evidence currently looks roughly like this:

Potassium citrate → strongest clinical evidence for appropriate calcium-stone patients with low urinary citrate.
Lemon/lemonade → reasonable human evidence that it can increase urinary citrate and/or urine volume, although results vary between studies.
Apple cider vinegar → interesting preliminary evidence, but not established treatment, and there isn't good clinical evidence that drinking it directly dissolves an existing calcium oxalate stone.

Also, the chemistry is important: kidney stones are not the same material as the white calcium carbonate scale inside a kettle. Most calcium kidney stones are calcium oxalate, so the observation that vinegar quickly removes kettle scale cannot be directly applied to kidney stones in the body.

Foods generally don't contain large amounts of potassium citrate itself in the same form as the prescription/supplement. Instead, many fruits and vegetables provide potassium plus citrate/citric acid or other alkali precursors, which can support higher urinary citrate.

For kidney-stone prevention, some particularly useful choices are:

Food	Why it may help
🍋 Lemon / lemon juice	Very high in citrate/citric acid; one of the best-studied dietary sources
🍋‍🟩 Lime	High in citrate
🍊 Orange / orange juice	Provides both potassium and citrate
🍈 Grapefruit	Contains citrate and potassium*
🍈 Melons	Good potassium source and high water content
🥑 Avocado	Very high potassium, but not especially high in citrate
🥔 Potato	High potassium
🍌 Banana	Good potassium source
🥝 Kiwi	Potassium plus organic acids
🍅 Tomato	Potassium and high water content
🥬 Leafy vegetables	Potassium and dietary alkali

Lemon is especially interesting for calcium-stone prevention because of its citrate content. Human studies have shown that lemon juice/lemonade can increase urinary citrate in at least some people with low urinary citrate.

One important distinction: eating a banana or potato gives you potassium, but that's not equivalent to taking potassium citrate. Prescription potassium citrate delivers potassium specifically combined with citrate and produces a more predictable effect on urinary citrate and urine pH.

For someone who forms calcium oxalate stones, a practical dietary approach is usually plenty of water + citrus (especially lemon) + normal dietary calcium + lower sodium, rather than simply trying to eat the highest-potassium foods.

* Grapefruit can interact with several medications, so it isn't appropriate for everyone.