You are correct: the verb italicize is limited to printing in italics. (Curiously, non-italic typeface is referred to as Roman). Cursive, while also slanted, is characterized by connected letters.
The more general verb is
Transitive. To make Italian in character or style.
She Italianised her Christian name.
The Hall was new wainscoted and thoroughly Italianized. OED
Make Italian in form or character.
Some migrated to Italy where they Italianized their names Lexico
1: To print in italics or underscore with a single line
The microphone italicizes every curdled top note m-w
Even Joan Sutherland was urged to Italianize her name. Jack
Gottlieb; Funny, It Doesn't Sound Jewish (2004)
Durward Lyall is a Scotchman by birth, who, we are informed, "will
probably Italianize his name, and, on the completion of his
studies at Milan, make his public debût on the stage." May we suggest
that Mr. Lyall has a very good singing name of his own, which there is
no need to "Italianize?" The Musical Standard, Jan. 30, 1875
After the establishment of the new governorship, the Italians began a
drive to Italianize the annexed territories, as they had worked to
Italianize Istria, the Slovene Littoral, and other South Slavic areas acquired after the First World War. Jozo Tomasevich; War and
Revolution in Yugoslavia, 1941-1945 (2002)
The difference was the Fiume's 1919 law allowed Fiumians to
voluntarily Italianize their family names, without any standards, guidelines, or requirements except that the new surnames
had to be "strictly Italian" and could not replicate those of
patrician or famous families. D. K. Reill; The Fiume Crisis (2020)
There is also a much less common verb:
To render Italian; to give an Italian character to; to Italianize.
Soft and voluptuous measures Italianating the rude tongues of the preceding generation. OED
At the same time, the Italian could still exert a certain fascination. In medical matters it was enough to call oneself 'a Padua doctor' to
secure success. 'Tis but Italianating my name, garb, language, and
habit, and the Segnior Quackquinto may practice', says the English
mountebank in Francis Quarles's The Virgin Widow, first performed in
the 1640s. David Gentilcore; Medical Charlatanism in Early Modern
In Canton, Larmour built Grace Church, its adjoining parsonage, and
the Odd Fellows Hall, and he may have been responsible for
Italianating the David Fulton House with a three-story tower flanked by twin front-facing gables, created around 1855. Patti Carr
Black; Art in Mississippi, 1720-1980