Leaving Sweden with a different passport?

Phoog’s answer is correct for normal times. However, in August 2020 Europe might still have some kinds of Covid-19 type restrictions. As of May, if that is the case and which restrictions will be in place in August is anyone’s guess.

In normal times, you would only have to establish your identity with the airline at some point during check in or boarding for which the Argentinian passport should be enough – as would a Spanish national ID card (it is not the airline’s duty to check whether you are legally in the country based on stamps and visas afaik). An expired Spanish passport might also work. You would not encounter any immigration officers along the way and airport security would only need your boarding pass.

In Covid-19 times, any of the three countries you would be touching (Sweden, Denmark and Spain) may have installed temporary border controls conducted by police or immigration officers. If you do, your Argentinian passport will likely raise the wrong kind of questions such as ‘Where is the entry stamp?’, ‘How long have you been in Sweden?’ and ‘Where is your visa?’ You will want to avoid that at all costs. This only leaves the Spanish passport or a Spanish ID card.

As crossing an internal Schengen border still requires carrying a valid ID document (even if it doesn’t have to be shown), an expired Spanish passport might raise questions as it does not unambiguously show Spanish citizenship. However, officers might still exercise goodwill, especially if the passport is recently expired and since you are using it to travel back to Spain (that does not work for the return leg, though).

I personally would not feel safe using any of these almost options. There are still over two months until August which should be sufficient to renew your Spanish passport. In the absence of a Spanish ID card (and if it is not possible to acquire one in time), renewing the passport would be my method of choice.