Sieh mal, der schwarze Laptop.
but
Du siehst den schwarzen Laptop.
This sentence is an imperative (command) in German, asking someone to look at something specific, in this case, "DER SCHWARZE LAPTOP." Here, "DER SCHWARZE LAPTOP" is in the nominative case because it is the subject of the command. It's similar to saying in English, "Look, the black laptop."
I agree with Janka that the part with the nominative is not a full clause, but I don't think we can explain all examples as ellipsis (that the verb has not been pronounced). Rather, the nominative is the normal case for free-standing items, and in an exclamation you can simply name a thing in order to draw attention to it. True, in many examples you could add the predicate "da ist":
But a peculiarity is that you can use a definite article because you are talking about something that is present in the situation (hence identifiable), while the predicate "...ist da" would need an indefinite because it is a grammatical structure that introduces new things. Compare:
For some reason, the last sentence could only be used if the swan has already been talked about. The first example, in contrast, can be an exclamation to direct attention to something new (but visible).
So the point of the construction in the question is that mentioning a thing (in nominative case) has the function to make someone look at it, but it is not a grammatical object of the verb "look" and not necessarily the subject of a missing verb; it is a free-standing noun phrase.
The part after the comma is not a complete clause as it has no verb. This is the full version:
Sieh mal, (da ist) der schwarze Laptop.
So this der schwarze Laptop is the subject of the second main clause. Hence it must be put in nominative case. It is not the subject of the first main clause. In fact, you could put this as two separate sentences as well:
Sieh mal! (Da ist) der schwarze Laptop.
