Bone fractures are very common injuries that pose a significant burden to healthcare systems worldwide due to their high incidence, impact on quality of life, and treatment costs. Traditional fixation implants, typically made of stainless steel or titanium, are effective but can cause complications such as inflammation due to foreign body responses, and alterations of physiological load transfer mechanisms. Bio-absorbable implants based on Magnesium-Gadolinium (MgGd) alloys, have recently emerged as a promising alternative, since their controlled degradation fosters bone regeneration while maintaining the necessary mechanical support, reducing the need for surgical removal. Computational modeling plays a crucial role in this context, enabling the development of comprehensive, patient-specific tools to optimize surgical strategies. However, scientific literature on the in silico modeling of these devices still remains limited to few, very specific cases. In this work, we present a general finite element framework to simulate the chemo-mechanical behavior of degrading MgGd-based screws. A thermodynamically consistent phase-field approach is employed to model the dissolution process, effectively capturing morphological variations of the screw over time. Model predictions are validated against available experimental evidence, showing a very good agreement. A parametric analysis has allowed to identify the effect of Gadolinium content on the alloy degradation kinetics, highlighting the most influential factors. Furthermore, fatigue analyses are conducted to evaluate the progressive mechanical weakening of the degrading implant under cyclic loading conditions. The proposed results provide valuable insights into the long-term behavior of bio-absorbable MgGd implants and represent a first step towards the development of predictive computational tools tailored to patient-specific needs.