There is some pictorial evidence supporting the idea that ships did indeed have painted things on their sails, crosses more often than not.

From Portugal there is the famous Livro das Armadas, which is full of depictions of different types of ships: caravels, cogs, urks, galleons, whatever. In most of the ships, you can see the cross of the Order of Christ, which is a very well known Portuguese symbol, painted on the sails. Some examples:

Joao Serrao's caravel with the cross

Pedro Alvares Cabral's fleet, with all of his ships showing the cross on the sails.

Luis Pires' carrack, also sporting the cross

The famous map of Domingos Teixeira from 1573 has over half a dozen Portuguese ships depicted on it, and all of them have a red cross painted on the white sails.

Piri Reis' map from 1513 shows a number of ships along the Atlantic Ocean, and looking closely one can see that some of them have some elements painted on the sails, painted in red. The use of red paint makes sense, as it would be easy to see against the sail and even against the sky, making it highly distinguishable (that's also why Spain's flag is red and yellow, though it is from 1785).

Here is the Piri Reis' map: https://upload.wikimedia.org/wikipedia/commons/7/70/Piri_reis_world_map_01.jpg

The presence of crosses painted on the sails was something well planted in the minds of the populace. In the Basel edition from 1494 of Columbus' letter anouncing the discovery of the caribbean islands, the ship depicted there has a cross on the sail. Of course, the engraver would not have actually seen Columbus' ship, but on his mind, ships had to have a cross on the sail.

https://en.wikipedia.org/wiki/File:Columbus_Letter_(Basel_1493)_Illustration_2.jpg

There is plenty of evidence on the contrary too, with lots of ships depicted with plain white sails, like the Catalan Atlas from 1375, which does not show ships with painted sails. By the Canary islands you can see a galley with its sail deployed, but no cross or anything on it.

https://upload.wikimedia.org/wikipedia/commons/b/b2/1375_Atlas_Catalan%2C_Europe_01.jpg

Pedro de Medina's "Arte de Navegar" from 1546 has a number of ships depicted on page C IIJ (title page of book II), but none of them have anything painted on the sails.

http://biblioteca.galiciana.gal/es/consulta/registro.do?id=9489

Plenty of paintings, engravings, etchings, etc that could have come in handy are not that useful, as they depict the ships with the sails taken up instead of deployed, which is a pity. The virgin of the navigators, by Alejo Fernández, is a brilliant painting, but has the problems I have already mentioned.

From the surviving iconography, we can say that some sails were painted, but it was mostly a Portuguese custom. Maybe u/terminus-trantor knows more on the matter.