When did western doctors first start predicting prenatal diseases and disabilities?

As currently phrased this question is only answerable with the trope of it's older than dirt as these kind of predictions and advice are of absolutely primary concern to all fathers and mothers since antiquity. While some assess even Imhotep, Hippokrates or Galen as the "fathers of modern medicine", others qualify these works and methods as way too pre-scientific for "modern medicine". Nevertheless, both Hippokrates and Galen, among others, describe, diagnose, supervise pregnant women. The explanations given, however, are mostly no longer satisfactory for modern readers and their understandinng of "diagnosis".

Perinatology is a new medical discipline, originating in the 1960s, and aiming to prevent, diagnose and treat problems to the mother, fetus and neonate, prior, during and after delivery. As above issues, applying to perinatology, are diachronically present, it is of interest to explore how they were previously viewed and coped with, particularly in ancient times.
Ancient Greece is considered the cradle of Western medicine. Thus, the reasonable question arises, what did Ancient Greeks think about unusual conditions, encountered around delivery, like premature birth, small for gestational age babies, twin pregnancy, as well as inability to give birth v***nally, and thus to apply what we nowadays call an emergency cesarean section (CS).
_{Ariadne Malamitsi-Puchner & Despina D. Briana: "Ancient Greek view of perinatal risk issues: from myth to reality", Journal of Pediatric and Neonatal Individualized Medicine 2017;6(2):e060223 doi: 10.7363/060223.}

If this is framed within our current understanding of modern, western medicine, then indeed our current understanding of amniocentesis seems to be the key development to enable that specific field of "prenatal diagnostics".

This started as a procedure to be considered for routine diagnostics in the 1930s and became an early and widespread application of this procedure in the 1950s already: for blood type mismatches and resulting consequences.

But that had an antecendent as well:

One of the key developments that helped to de ne morbid heredity as a distinct source of birth defects was a radical transformation of the meaning of monstrous births. In the early modern period, scientists and physicians, as well as religious authorities and laypeople, were fascinated by births of severely deformed children. Such births were given numerous, sometimes contradictory, meanings. Monstrous births were both repulsive and marvelous. They were seen as a demonstration of divine wrath and the endless power of divine creation but also as natural phenomena that displayed the playfulness of nature and its mistakes, the vagaries of chance, and the in nite creativity of the living world and might of maternal imagination.

In the seventeenth century, the investigation of monstrous births, human and animal, gradually shifted to the explanation of such births as natural phenomena. In the mid- to late eighteenth century, British and French physicians viewed “monsters” as clarifying counterexamples to normal embryological development.

During the second half of the nineteenth century, interest in abnormal births favored the development of experimental embryology and new approaches to the study of pregnancy in humans. Edinburgh gynecologist John William Ballantyne (1861–1923) began his career as a specialist in teratology. Later, he moved from the investigation of “monsters” to finding ways to prevent birth defects. Ballantyne’s major contributions to the subject were his books Diseases and Deformities of the Foetus (1892–1895) and Manual of Antenatal Pathology and Hygiene of the Foetus (1902–1904), which were dedicated to the study of fetal abnormalities. Diseases and Deformities of the Foetus focuses on the description and classification of fetal anomalies. Manual of Antenatal Pathology and Hygiene of the Foetus was written with the explicit goal to shed light on the advantages of preventing the birth of impaired children. Before Ballantyne, embryology was seen exclusively as a domain of fundamental biology. The practitioners’ interest in pregnancy was mainly limited to childbirth accidents and did not include the unborn child. Thanks to Ballantyne’s studies, medical practitioners became interested in life before birth. Gynecologists, Ballantyne argued, have a unique advantage as students of antenatal development. They have the ability to observe both pregnant women and their offspring, and can thus unravel correlations between events that precede birth and their consequences.

Ballantyne’s Diseases and Deformities of the Foetus is a book on teratology with a practical twist. The book starts with a very long historical section, grounded in sources from antiquity, the middle ages, the renaissance, and the eighteenth and nineteenth centuries. It is dedicated to physician Cesare Taru from Bologna, the author of Storia Della Teratologia (published in 1889), and biologist Camille Dareste, director of the teratological laboratory at École des Hautes Études in Paris and author of Recherches sur la production artificielle des monstruosités (published in 1877; a second, enlarged edition of this book was published in 1891). Diseases and Deformities of the Foetus is richly illustrated with line drawings and is organized according to an anatomical schema: each chapter is dedicated to the description of a specific pathological condition. Fetal anomalies, Ballantyne explained, are not mere curiosities: they are of utmost importance to the clinician. Until recently, physicians were not interested in the fetus because they attributed low value to fetal life and health. Such a view is decidedly erroneous. Fetal malformations are the main cause of high mortality in the neonatal period. Moreover, many such malformations can be prevented. Fetal problems that lead to miscarriages, stillbirths, and birth of sickly new-borns are more often observed in mothers who suffer from poor physical or mental health. Certain paternal diseases, such as syphilis, tuberculosis, alcoholism, and cancer, can also diminish the quality of sperm, thereby affecting fetal health. The diffusion of the use of proper hygiene reduced mortality in adults; it is a doctor’s duty to promote the application of similar rules during pregnancy.

Diseases and Deformities of the Foetus was largely a treatise of teratology. In the Manual of Antenatal Pathology and Hygiene of the Foetus, Ballantyne’s focus shifted from a description of “monstrosities” (i.e., an abnormal form) to the study of the pathologies of the fetus (i.e., an abnormal function). According to Ballantyne, we do not know the precise cause of some fetal and newborn pathologies, such as inborn malformations of the digestive and nervous systems, but we do know the cause of many other pathologies: for example, poor maternal health. Infectious diseases such as syphilis and tuberculosis, and chronic conditions such as diabetes or cancer directly hamper the development of the fetus. The same is true for poisons: lead, mercury, arsenic, alcohol, and tobacco. It is difficult to prove the direct effect of smoking tobacco on the fetus, but the high mortality rate of children born to women who work in the tobacco industry provides strong indirect evidence of this substance’s harm. Ballantyne also noted that women who drink heavily during pregnancy often give birth to malformed children, probably because alcohol afects embryonic or fetal development. Since the fetus is not isolated from the maternal body and is therefore strongly afected by the maternal environment, it is a doctor’s responsibility to educate pregnant women about antenatal hygiene. Such an education will prevent many birth defects. Pregnant women should see a doctor on a regular basis, and those with severe health problems should be treated in “prematurity hospitals” to reduce risks for the newborn.

On the evolution of Ballantyne’s ideas, see Salim Al-Gailani, “Teratology and the Clinic: Monsters, Obstetrics and the Making of Antenatal Life in Edinburgh, c. 1900” (PhD diss., University of Cambridge, 2010). Ballantyne continues to be seen as the “founding father” of teratology. J. Bruce Beckwith and Ronald J. Lemire, “John William Ballantyne—a Biographical Sketch (1861–1923),” Teratology 1 (1968): 1–3.

_{Ilana Löwy: "Imperfect Pregnancies. A History of Birth Defects and Prenatal Diagnosis", Johns Hopkins University Press: Baltimore, 2017.}