We develop a self-consistent first-principles framework for determining the screened Coulomb interaction strength $\left(U\right)$ based on constrained dynamical mean-field theory (cDMFT). Unlike conventional approaches, this method incorporates essential vertex corrections within the same embedded-DMFT formalism used for the electronic structure calculation. Using the cDMFT-derived interaction strengths as input to embedded DMFT yields spectral functions in excellent agreement with photoemission experiments across a wide range of materials, spanning $3d$ to $5d$ transition-metal compounds, including correlated metals, Mott insulators, altermagnets, and unconventional superconductors. This unified many-body framework establishes a systematic first-principles route for determining interaction strengths in correlated materials and substantially enhances the predictive power of $\mathrm{DFT}+\mathrm{DMFT}$ and its extensions.

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• Electronic structure
• Strongly correlated systems
• Dynamical mean field theory
• First-principles calculations