Obstetric ultrasound is a fundamental ingredient of modern prenatal care with many applications including accurate dating of a pregnancy, identifying pregnancy-related complications, and diagnosis of fetal abnormalities. However, despite its many benefits, two factors currently prevent wide-scale uptake of this technology for point-of-care clinical decision-making in low- and middle-income country (LMIC) settings. First, there is a steep learning curve for scan proficiency, and second, there has been a lack of easy-to-use, affordable, and portable ultrasound devices. We introduce a framework toward addressing these barriers, enabled by recent advances in machine learning applied to medical imaging. The framework is designed to be realizable as a point-of-care ultrasound (POCUS) solution with an affordable wireless ultrasound probe, a smartphone or tablet, and automated machine-learning-based image processing. Specifically, we propose a machine-learning-based algorithm pipeline designed to automatically estimate the gestational age of a fetus from a short fetal ultrasound scan. We present proof-of-concept evaluation of accuracy of the key image analysis algorithms for automatic head transcerebellar plane detection, automatic transcerebellar diameter measurement, and estimation of gestational age on conventional ultrasound data simulating the POCUS task and discuss next steps toward translation via a first application on clinical ultrasound video from a low-cost ultrasound probe.