What do you call a phrase that doesn't need to be completed?

Solution 1:

In addition to what has already been said, this type of phrase might also be considered a Synecdoche:

a word or phrase which uses a part of something to represent its whole. Some general examples include using "suits" to refer to businessmen, "threads" to refer to clothing, and "stick shift" or "stick" to refer to the type of manual transmission of a car or even the car itself (e.g., "Can you drive a stick shift?").

Examples I can think of that are similar to yours include:

"Life is like a box of chocolates" to refer to the longer quote from Forrest Gump,
"Death and taxes" to refer to "The only two certainties in life are death and taxes",

and possibly

"Jack of all trades" to refer to "Jack of all trades, master of none" or "Blood of the covenant" to refer to "Blood of the covenant is thicker than the water of the womb".

Solution 2:

Could it be the word ellipsis?

According to Oxford Dictionaries :

The act of leaving out a word or words from a sentence deliberately, when the meaning can be understood without them.
Three dots (…) used to show that a word or words have been left out.

Solution 3:

I would call this an implication, or even an implied cliche.

Google defines an implication as "the conclusion that can be drawn from something, although it is not explicitly stated", and a cliche as "a phrase or opinion that is overused..."

Solution 4:

Elliptical phrase or elliptical clause seems to be the name

elliptical clause: A clause in which something is omitted, usually because it is understood. In the sentence “If in doubt, check the manual.” “If in doubt” is an elliptical clause, with words such as “you are” omitted.

In using such phrases you may be speaking 'elliptically'.

elliptic adj. 1. Of, relating to, or having the shape of an ellipse.
2. Containing or characterized by ellipsis.
3. a. Of or relating to extreme economy of oral or written expression.
b. Marked by deliberate obscurity of style or expression.

Solution 5:

Phrases so popular that they need not be completed to be understood are a kind of common currency, perhaps (i.e. something that a lot of people know about and talk about).

The name for phrases where only a part is used (with the rest understood) is ellipsis.