Learn Russian: Врач смотрит живот. - The doctor examines the stomach.

While смотреть usually means "to look" or "to watch", it is the most common everyday verb used when a doctor checks a patient. Just like in English you might say "the doctor is taking a look at my stomach," in Russian, they simply use смотрит.

Живот is the direct object here. Why didn't its ending change?

Because живот is a masculine, inanimate (non-living) noun. In Russian, when a masculine noun is inanimate, its direct object form (the Accusative case) looks exactly the same as its dictionary form. If the doctor were examining a feminine noun, the ending would change.

The dictionary form is смотреть. Why does it conjugate as смотрит with an "и"?

The verb смотреть is a famous exception! Even though it ends in -еть, it conjugates like verbs ending in -ить (the second conjugation group). That is why it takes the -и- endings, making it он/она смотрит instead of using an -е-.

The English translation says "the stomach". Why isn't there a word for "the" or "his"?

Russian does not have articles like "a" or "the". Additionally, Russians usually leave out possessive words like "my" or "his" when talking about body parts if it is obvious from the context. It is just understood that the doctor is examining the patient's stomach.