Calligraphic painting follows on from biomorphic painting and there is a fine line between the two at the beginning of the 20th Century. Paul Klee's They're biting, makes an interesting comparison with one of Archile Gorky's paintings, which features lamps and what looks like a ceiling fan. The forms grow from biomorphic shapes, and likewise with Klee, although one can identify the fish etc. the shapes are almost biomorphic. Klee's painting Cosmic Flora of 1923 hints at callibraphic marks.
Stuart Davis in 1924 was probably the first to introduce text and advertising with his Odol painting, wich eventually led to Pop art. Charles Demuth was also painting at this time and in 1928 painted I Saw the Figure 5 in Gold. Even earlier, Joseph Stella, in his Battle of Lights, Coney Island painted in 1914 introduced calligraphy for the name of the Park.
Franz Klein painted large black paintings which featured calligraphic shapes which are almost Eastern in influence. He uses calligraphic marks quite differently from other artists by enlarging them and painting them in black as the only subject matter of the painting.
Willem de Kooning also used a mixture of biomorphic and calligraphic motifs in his work which is identified as action painting because of these interesting gestural marks which have been worked at speed.
Adolph Gottlieb's painting, The Alchemist, uses symbols and pseudo textural marks in a grid construction, and in this particular case it relates to the subject of the painting as the symbols are scientific in character.
Jasper John's uses calligraphic text in his painting Periscope, and in his readymades. He also uses text for Savarin, the name of the Toffee tin, as well as beer cans and the figures in work that alludes to dollar bills. Any Warhol, uses text on soup tins in the same way, and of course text appears when using collage, particularly Richard Hamilton.
It's difficult to define Jackson Pollock's work as calligraphic, I would say more biomorphic, and in this way he uses paint in a similar way to Willem de Kooning, and later Frank Stella.
Robert Cottingham's photorealistic work features text but in an illustrative way, not as a motif, in much the same way that Roy Lichtenstein includes it as " speech bubble" text.
Cy Twombly uses text a lot in his paintings and of course many Conceptual Artists do to, particularly with Blinkey Palermo's Latitude and Date paintings.