Your question was,

I can't seriously be held responsible for this, right?

Based on what you said here,

I never signed anything saying that I'd owe them money if I quit - I have a copy of the signed employee agreement with nothing anywhere close to that in it. The AutoCAD license was installed on a work computer that never went home with me and I returned everything they gave me on my last day, down to the free shirt and water bottle.

Then I think the answer is pretty clearly, no. You can't be held responsible. It is sometimes the case that employment contracts include terms for reimbursement of specific expenses or repayment of specific types of benefits (ie an employer that pays for you to go get your Masters degree may have terms in your contract stating that you need to pay them back if you quit within a certain time frame), but that is apparently not the case for you.

It seems like your best course of action is to ignore the request. Any response you make may be seen by them as grounds to continue to push you. If they escalate, it would be appropriate for you to do so as well (ie get a lawyer involved). Otherwise, I would not acknowledge their request in any way.

Oh, keep this, and your signed agreement in a very secure location.

Depending on your location, you may be needing them for a civil suit. Your employer will probably end up wishing that it only cost them $7000.

FWIW: I am inferring the U.S. because of the currency denomination. You might want to specify a location tag.

I can't seriously be held responsible for this, right?

Right. It's a bluff to see if you would pay anyway.

Should I respond that I'm not paying, or just ignore this?

I would respond once and only once with something like:

"Since my employment agreement said nothing about being required to reimburse any training expenses if I left, I will not do so.

The AutoCAD license was installed and remains on a company computer. The computer was returned when I left. I am not responsible for the cost of this license and will not provide any reimbursement."

Then I would not respond to subsequent letters, although I would keep copies of everything.

In the unlikely event you are sued, make sure you have all the relevant facts and copies of all correspondence. I have a hard time imagining they would waste their time doing so, and an even harder time imagining they could win.

You don't have to do a thing, but there are ways to handle it. In some countries like the US, it may be better to stay silent or ask a lawyer (which costs!), because the law can be very harsh and expensive if done wrong. If you're more sure of writing, then you can probably sort this out yourself.

I'll sum up what I'd expect is a good approach, but you may choose to do nothing and "wait and see", or ask a lawyer, in some places.

How the law sees it

When a business enters a contract, it is assumed that they know what they are doing. A private individual often gets more protection, because they often don't. (Which is why consumer and employment law often strongly favour the consumer/employee).

Your company knew when they employed you, that they wanted certain conditions as part of that employment. They knew you could leave on certain terms, and accepted that. In fact, they probably told you (dictated to you?) what those terms would be. They actually chose them, themselves.

When they bought you software, or trained you, they knew you had that right to leave. If they wanted to make training or purchases conditional on you staying, or chargeable if you didn't, then they had every chance in the world to say if that mattered to them. They could have written it in the original contract (it was only 4 months ago so they surely had some idea of what they'd do!). They could have asked you at the time to sign an additional agreement that you wouldn't leave for a year or would repay if you did, in return for this stuff they were about to do. They didn't do any of that, and they had the choice, the ability, and the awareness to do so. So they can't now look to you, to cover up their failure. They knew they had given you a right to leave without repaying, when they did all that, so they can hardly expect you to voluntarily repay if you take them up on the rights they signed with you.

Put differently, every contract carries risk. If you want to not have a risk, you don't agree to contracts that have the risks in them, that you aren't willing to agree to - much less author those contracts. They didn't want to expose themselves to a risk of investing in someone, only to have them leave after 4 months? Then they shouldn't have signed a contract by which they voluntarily put themselves in that position with their employee.

A second way a court will look at it, for the license at least, is to ask if the company actually lost anything. Presumably they hired you because they needed someone to do work that would involve AutoCAD. They surely didn't buy it as a luxury! So presumably that work still exists. So they'll be able to use the license for their next hire.

After all, every employee who walks, could be sued if that basis were valid. You walked out after a week, and your job was inputting data from invoices? Well, we had to take time to recruit you, train you, get you a desk and chair, pay for the floor space you took up, pay for a computer and those internal support time you needed from IT to get it working.....

No. Just no. It doesn't work that way.

What to do

As I said, you may choose to wait, or get advice. But if you intended to write, you would want to write in a way that has a high chance of shutting them up :)

I tend to write strongly, and you may not feel comfortable doing so (and it may in fact be very unwise in some countries or situations to do so, or to not get legal advice). I'm also used to UK laws and legal frameworks where writing this way is a pretty safe bet, which might not apply where you are.

But if you did want to write or handle it yourself, and felt it was safe and you were confident to do so, this is roughly what I'd say, so you can pick the key points and use them as you feel best.

Something like this:

"Dear X,

I am in receipt of your letter of Xx March 2019, which looks to me like a simple attempt to extort money without good grounds.

I was employed under a contract. That contract was authored by Y Inc. itself, and the terms were chosen by and agreeable to Y Inc. The contract stated my rights to leave, and what happens when I leave. I attach a copy for your reference.

If you believe that the contract states that I must pay any sums on training or software you bought as an employer, I would appreciate you citing the paragraphs that contain those agreements. You will find no such agreement, of course, which means I do not expect to be troubled by you again on this spurious claim.

Having set out the contractual situation, I now consider this matter permanently closed. I reserve all of my rights to consider any further attempt to seek payment of this kind, or other adverse action taken in lieu, as likely to be an extortion attempt or other illegal action, and reserve my rights to seek any combination of legal advice, compensation or to report it as a police matter. I may also choose to ignore unfounded claims, if any.

Kindly do not write again.

Signed, Z."

[The "other adverse action" and "other illegal action" is in case they try to blacken your name or make harmful claims in revenge, which some might consider. But I haven't said that explicitly since we don't need to give them foolish ideas. Also note the thin line in defamation law - it is usually very safe to say "It looks to me like X", because that's a true statement of honestly held opinion, but it's much less safe to say "it is X", which claims a true statement absolutely]

If you have no documentation stating that you owe them this money (in your contract/employment agreement/etc), then tell them as such: "I never agreed to this, go shove it" (depending on how you feel about this company, I might be tempted to use those exact words) and do not respond to any further communique. If they continue to "shout into the void", as it were, you may want to contact a lawyer regarding harassment.

If you feel like being extra cheeky though, you may want to suggest, at least, if you have to pay $1500 for the AutoCAD subscription, then it should be your property. Issue them a notice to have your AutoCAD license transferred to you on threat of a lawsuit for stolen property, and see what they say. Of course, this is likely to not work out in the way you hope it will, so only do it if you feel like being cheeky and having some fun with a possible potential downside cost down the road. I don't actually recommend doing this, I just think it would be funny to hear their response.