Learn Finnish: Lääkäri vakavoitui, kun potilas masentui. - The doctor became serious when the patient got depressed.

Masentui is the past tense of masentua (to get depressed). It is derived by adding the -ntua suffix to the root of the transitive verb masentaa (to depress / to bring someone down). This suffix creates an inchoative verb, which means it expresses undergoing a change of state on your own, rather than doing the action to someone else.

Vakavoitui also expresses a change of state ("became serious"). Why doesn't it use the -ntua suffix?

While many adjectives use -ntua to mean "becoming [adjective]" (like pieni becoming pienentyä), the adjective vakava (serious) uses a different inchoative suffix, -oitua, resulting in vakavoitua. Finnish has a few different suffixes for verbs of becoming, and which one to use depends on the base word, though they achieve the exact same grammatical purpose in this sentence.

What happens to the stem when we put these -tua verbs into the past tense?

Because both vakavoitua and masentua are Type 1 verbs ending in -ua, forming the past tense is very straightforward. You take the infinitive, drop the final -a, and add the past tense marker -i. This creates a natural -ui diphthong at the end, resulting in the forms vakavoitui and masentui.

Why are the subjects lääkäri and potilas in the nominative case? Don't changes of state sometimes require the translative case?