No, oxidation is a broader group of reactions than simply something reacting with oxygen. See the explanation...
In its most general sense, oxidation is the loss of one or more electrons by an substance. While it can (and often does) happen that oxygen is the chemical that takes the electrons from this substance, there are many other such "oxidizing agents". This is what we call a substance that has the ability to acquire electrons (thereby causing the oxidation of that substance).
For example, each of the halogens is an oxidizing agent and can cause oxidation. This is what makes them effective as bleaches and as antiseptics and bactericides.
So, a redox reaction is the name we give to any process that occurs because one substance has been a donor of electrons (it is oxidized) while another substance is an acquirer of electrons (it reduces).
Oxygen simply has a very strong tendency to act as the second of these substances - acquiring electrons - and causing oxidation. However, it is not even the strongest of all oxidizing agents. That distinction goes to fluorine gas.