Dietary nitrate is reduced to nitrite by oral microbiota, and nitrite is subsequently converted to nitric oxide in systemic circulation.

Study: Ageing modifies the oral microbiome, nitric oxide bioavailability and vascular responses to dietary nitrate supplementation.

The study shows that when people eat nitrate-rich foods like beetroot, bacteria in their mouth turn it into nitrite, which then becomes nitric oxide in the blood — helping to lower blood pressure. This proves the process described in the claim actually happens in humans.

Study: Nitrate reduction capacity of the oral microbiota is impaired in periodontitis: potential implications for systemic nitric oxide availability

The study shows that good bacteria in your mouth turn nitrate from veggies into nitrite, which your body uses to make nitric oxide. If you have gum disease, these bacteria are fewer, so less nitrite is made — meaning the process doesn’t work as well.

Study: What's in Your Beet Juice? Nitrate and Nitrite Content of Beet Juice Products Marketed to Athletes.

This study shows that beet juice has lots of nitrate but very little nitrite, which means your mouth bacteria probably turn the nitrate into nitrite, and then your body turns that into nitric oxide—exactly what the claim says.

Study: The chemistry of the nitrate–nitrite–nitric oxide pathway: regulating muscle oxygenation and exercise performance

The study shows that when you eat foods like spinach or beets, bacteria in your mouth turn the nitrate in them into nitrite, and then your blood turns that nitrite into nitric oxide, which helps your muscles get more oxygen. This is exactly what the claim says.