Exactly so! What you have in this case (E, B, and e) is not a chord, it is an interval of a perfect 5th with the root doubled. It's neither major nor minor. You need a G or a G# to pin it down as either one or the other.
If you have an E, a G#, and another e, you also have an interval -- a major 3rd with the root doubled. But you are stongly implying an E major chord.
I was taught that if you had to leave a note out for some reason (maintaining correct voice-leading in class harmony exercises, for example), leave out the 5th. Root and 5th interval is ambiguous. Root and 3rd interval implies a chord. But it's still not a chord.