While certain forms of zinc may be important for immune responses to pathogens, it is entirely unknown whether oral zinc supplementation would have any antiviral activity against the COVID-19 virus, SARS-CoV-2, and it would certainly not be safe to test this by ingesting calamine lotion.
Zinc is an essential trace element that is necessary for human health and must be consumed regularly through a healthy diet (in red meat, poultry, beans, nuts, some seafood, whole grains, etc.). Zinc is important for normal growth and development as well as immune responses to pathogens and studies have shown that populations that are deficient in zinc are at risk of viral infections such as HIV or hepatitis C.
Clinical studies have yielded mixed results on whether oral zinc supplements (like those found at the grocery store or pharmacy) can improve the clinical outcome of a viral infection. These studies have used a wide range of doses and formulations (i.e. zinc aspartate, zinc gluconate, zinc sulfate) but none has ever used zinc oxide, which is an ingredient in the over-the-counter anti-itch cream calamine lotion, or pyrithione zinc, an antimicrobial compound found in shampoos and face creams.
Calamine lotion and beauty products containing pyrithione zinc are only intended for topical application and should NOT be swallowed. In case of accidental ingestion, you should seek medical assistance or contact a Poison Control Center immediately.