Probably yes, but at the DM’s option
First, I assume the DM rules that some weapon you are proficient in is “similar” to a weapon your “character is proficient with.” I have a hard time imagining how anyone could argue that this is not the case, since to do so would require claiming that the weapon is not an object “similar” to itself. But at any rate, the relevant rule is explicitly “at the DM’s option.”
Second, there is a grammar ambiguity in that rule, about what exactly the word and means in a particular place, and the interpretation of that and directly impacts whether or not you get proficiency. Since this is in the rule about the DM’s options here, I’m calling the ruling on that ambiguity to also be part of the DM’s option.
For what it’s worth, as DM I absolutely would allow proficiency here. Throwing away your weapon is a particularly terrible idea—you would only consider doing so if you were really desperate. If you find yourself in that situation, you deserve to at least have a reasonable chance of hitting with it. And I think it matches the rules text better.
Relevant rules quotes:
If a character [...] throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
This shows that when you throw a non-thrown-melee weapon, it is an “improvised thrown weapon.”
In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club.
This is a possibility for any improvised weapon. A non-thrown-melee weapon that gets thrown counts as a improvised thrown weapon, which is to say it is an improvised weapon, and since it actually is a weapon, it is certainly “similar” to the weapon it actually is.
At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
And this shows that in the case of an improvised weapon being similar to an actual weapon, which almost-certainly must be the case here, proficiency bonus applies.
Except, and this is an important bit, we have “[you] can use [it] as if it were that weapon and use [your] proficiency bonus,” here. That and is ambiguous: either this rule lists two separate things you “can” do, and they are independent, or this is listing one thing you can do, which consists of both parts together. That is, which of these is the correct parsing of the sentence?
- use [it] as if it were that weapon
- use your proficiency bonus,
- use [it] as if it were that weapon and use [your] proficiency bonus
The repeat of the verb use here points to these two clauses being two separate entries: you can use it as if it were that weapon, and also, you can use your proficiency bonus with it. But that is an exceedingly high level of legalistic specificity to expect of a D&D rule; we don’t generally assume that the rules are written that carefully. Which means that this is, arguably, ambiguous, and you (or more relevantly, the DM) can decide it means them as one thing you can do, and so you only get the proficiency bonus if you use the weapon-like object as that weapon. Throwing a greatsword would not be that.
(The other argument you could make is that, say, a greataxe is similar to a handaxe, which can be thrown, so if you throw a greataxe you are using it as if it were a handaxe, so your handaxe proficiency should apply. Also a plausible ruling for a DM to make. Certainly, if you bash someone with a bowstaff, arguing that it is similar to a club or quarterstaff seems quite reasonable.)
Irrelevant rules quotes
Just because there is other rules text here that seems to be confusing some folks, these are some rules that don’t actually apply to this situation.
An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object).
A weapon is not an object that bears no resemblance to a weapon. Therefore this sentence, in its entirety, does not apply, which is precisely why the next line, quoted above, says that throwing non-thrown-melee weapons “also” reduces their damage to 1d4.
If you use a weapon in a way that turns it into an improvised weapon—such as smacking someone with a bow—that weapon has none of its regular properties, unless the DM rules otherwise.
(Jeremy Crawford tweet)
We aren’t discussing a regular property of the weapon—by definition we are discussing a property the weapon doesn’t regularly have, since we are talking about the thrown property of a weapon that doesn’t have it—so the fact that a weapon doesn’t have its regular properties is irrelevant. Jeremy Crawford does not deny an improvised thrown weapon its irregular thrown property, which the rules explicitly grant it.