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I was helping a friend write a paper and came across a sentence which confused me.
The sentence was something along the lines of:

Horrifying consequences inflicted upon innocent people.

As soon as I read this, I knew it was a fragment, but could not describe why. I can distill it down to:

Consequences inflicted.

and from there it seems to follow that there is a subject (the consequences) and a past-tense verb (inflicted). In my mind, it is no different from the sentence:

Icicles melted.

I am fairly sure that the latter is a complete sentence whereas the former is not, but both seem to have a past-tense verb and a plural noun. I would love if somebody could shed a little more light on the situation. Thanks in advance.

It's a fragment because there is no required auxiliary verb.

For instance:

✔ Consequences were inflicted.

This is a valid passive sentence, along the same lines as:

✔ The window was broken.

In this sentence, broken is an adjective. (In the previous sentence, inflicted is acting as an adjective.)

In another construction, inflicted can be used without an auxiliary verb, but it requires an object.

✔ They inflicted themselves on him.

In your second sentence, the intransitive verb melted doesn't require an auxiliary verb:

Icicles melted.

Cars crashed.

They jumped.

Although extremely short, those are all still sentences with a subject and a validly constructed intransitive verb.

It’s not considered a sentence because it contains no subject (even implicitly, like an imperative). “Consequences” is grammatically a direct object of “inflicted.”

In formal standard written English, “Consequences inflicted” would not normally be written as a complete sentence. You would be more likely to see the phrase set off by a comma, perhaps, “Consequences inflicted, the mother left her son in his room.” If you did follow it with a full stop, it would indicate a pause for effect, as in “There would be repercussions. Consequences inflicted.”

Native speakers sometimes do say things similar to that. Most of the examples that come to mind are from the military: “Countdown initiated,” “Missile launched,” “Target acquired,” etc. The copula—the "is", "was" or "has been"—is implied but unstated. It’s a minimalistic way of speaking associated with situations where every second counts. Robots in science-fiction stories tend to speak this way, too.

After doing a little more research based on some helpful comments, I read about past participles at: https://webapps.towson.edu/ows/verbals.html

A verbal is a word formed from a verb but functioning as a different part of speech.
A participle is a verbal that functions as an adjective.

Past participles, usually ending in -ed or -en, are created from the form of a verb used with the verb to be as an auxiliary verb (passive voice).

For example: The windows were cracked by vandals.

In the example, like in mine, there is a main verb in the past participle form ("windows cracked by vandals"); however, this is not a complete sentence without the auxiliary verb of form "to be". In my example it would be a complete sentence if I added "were" before inflicted.

The difference between "consequences inflicted" and "icicles melted" is that inflicted is functioning as a past participle in this case.

"Inflict" is what's known as a transitive verb. A transitive verb is a verb that requires an object. Intransitive verbs don't have a subject (many verbs can be used both transitively or intransitively, but "inflicted" can only be transitive). When we write a subject and an intransitive verb, that can be complete sentence. "Melt" can function both transitively ("The sun melted the snow") or intransitively ("The snow melted"). Since "inflict" functions only transitively, "Consequences inflicted" is not a complete sentence.

Whether a verb is transitive or intransitive is often included in dictionary entries. For instance here there's

Definition of help (Entry 1 of 2)

transitive verb

1 : to give assistance or support to
help a child with homework

and then later on

intransitive verb

1 : to give assistance or support —often used with out
helps out with the housework

So "help" can function intransitively, hence "I helped!" is a complete sentence. On the other hand, if you look up "confuse", you will see only transitive definitions. So "It confused" would not be a complete sentence.

Furthermore, "inflicted" is probably not the past tense of "inflict", but the past participle, in which case it is functioning as an adjective, so there isn't any verb at all. For "inflict" to be the past tense rather than past participle, "consequences" would have to be the subject, but "consequences" makes more sense as the object than the subject.

Yes, a transitive verb always needs a direct object or the thought is incomplete and therefor not a sentence. Other examples of incomplete thoughts: John opened. The bishop slapped