No, that is not true. If GR said “the universe is expanding,” then Einstein would have said, “the universe is expanding” in 1916. But he specifically said the universe was “fixed”.
GR was specifically designed to keep the universe from either expanding or contracting.
All of this goes back to the question that Newton and others asked, “If the gravity of all astronomical bodies ‘pulls’ on each of the bodies, then why does the universe not collapse in on itself, due to all the gravitational pull?”
Newton suggested four possible solutions to that question: 1) maybe it is collapsing but we just don’t notice it; 2) maybe it is expanding but we just don’t notice it; 3) maybe the universe is infinite and all gravity ‘pulls’ cancel each other out in all directions; 4) maybe the whole universe is rotating.
Turns out that he was right about #2.
Turns out that Einstein was wrong and the universe was not “fixed”. He wrote a paper about this in 1932, in which he retracted his “curved space” idea and his cosmological constant. So, he not only didn’t “predict” the expansion, he was caught off-guard by it and had to change his GR theory to accommodate the expansion. Here are some excerpts from his 1932 paper:
”In a recent note in the Göttinger Nachrichten, Dr. O. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero.
There is no direct observational evidence for the curvature, the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor the value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing a curvature at all.
Although, therefore, the density corresponding to the assumption of zero curvature and to the coefficient of expansion may perhaps be on the high side, it certainly is of the correct order of magnitude, and we must conclude that at the present time it is possible to represent the facts without assuming a curvature of three-dimensional space. The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value.”
Full title of the paper: “On the Relation between the Expansion and the Mean Density of the Universe” Albert Einstein and Wilhelm de Sitter, Proceedings of the National Academy of Sciences 18, 213-214. Reproduced in “A Source Book in Astronomy and Astrophysics, 1900 – 1975”, published by Harvard University Press, Cambridge, MA, and London, England, 1979. Edited by Kenneth R. Lang and Owen Gingerich.
What started the new renewed interest in the possibility that the universe was “expanding” was Sliper’s work early in the 20th Century, and especially his 1917 paper, “A Spectrographic Investigation of Spiral Nebulae,” Proceedings of the American Philosophical Society 56, 403-409 (1917). He said in his paper that his spectrographic evidence revealed that the majority of spiral galaxies he examined were moving away from the Earth at high speeds, at speeds much faster than the speeds of the motions of the stars. So it was his work that told physicists as early as 1917 that the universe certainly was not “fixed” and that it seemed to be expanding. 10 years later Hubble confirmed Sliper’s findings. Five years after Hubble’s announcement, Einstein retracted his “universal curved space” idea and his “cosmological constant”.
You need to go back and read the original papers and original books. You can’t rely on information published in modern books and magazine articles, written by science writers who haven’t conducted the proper historical research. What they often do is merely copy two or three other recent articles they’ve read. I was in the magazine business, and I know how it works. Writers grab urban legends from other recent magazine articles and books, and they pass the legends along without conducting any in-depth research of their own.