Is combining two short complete sentences with a comma proper grammar?

These would require a semicolon. To join two independent clauses with a comma, you must use a coordinating conjunction. For example, you may say, “I have a goldfish. His name is Fred.” With a semi-colon, these two independent clauses become, “I have a goldfish; his name is Fred.” Note that the two clauses can be combined only if they are directly related to each other. Usually a semicolon would only be used if one sentence helps to clarify or enrich the other. In your original example, the second sentence explains what the word “it” referred to in the first. This would be an appropriate time to employ a semi-colon.

A comma could be used preceding a coordinating conjunction, as in, “I have a goldfish, and his name is Fred.” A common mnemonic device to remember coordinating conjunctions is FANBOYS. The letters stand for, “for, and, nor, but, or, yet, so.” These words explain clearly how the two independent thoughts are related to each other.

In relation to your original question, any of the above choices would be correct. Deciding which structure to use is mainly a stylistic choice. However, using clear and concise language when possible is always a wise strategy. It is often possible to rephrase your original ideas into one clear thought, such as, “I can see on his face that he feels dumb.”