Can 1 mole of H2 react partially with one mole of O2?
As you have said, you are studying stoichiometry at High School Level. From this I can guess, that you have probably not studied the concept of Limiting Reagent yet.
What is Limiting Reagent?

In a chemical reaction, the limiting reagent, also known as the &quot;limiting reactant&quot;, is the substance which is totally consumed when the chemical reaction is complete. The amount of product formed is limited by this reagent since the reaction cannot proceed further without it. The other reagents may be present in excess of the quantities required to react with the limiting reagent.

As per your question, $\ce{H2}$ is limiting reagent. Hence it will get totally consumed when reaction happens and 1/2 Moles of $\ce{O2}$ will be left.

1 mole of H2 and 1 mole of O2 won't make 1 mole of H2O and 1/2 mole O2.

1 mole of $\ce{H2}$ and 1 mole of $\ce{O2}$ will make 1 mole of $\ce{H2O}$ and 1/2 mole $\ce{O2}$ will be left.
$\ce{H2 + O2 -&gt; H2O + 1/2O2 (Left) }$
And here is an advice for you. If your teacher is saying that this will not happen at all, tell him to be sure that reactant in reaction are single molecules, not 1 mole.
Read this to be more clear on concept of limiting reagent.