A vast amount of enterprise and scientific knowledge remains locked in unstructured documents, while analytics and decision-making rely on relational databases. Bridging this gap is challenging because document extraction operates on local text spans, whereas relational databases enforce global semantics through schemas, keys, and integrity constraints. As a result, simply prompting large language models (LLMs) to extract tables often yields outputs that are hard to audit and inconsistent at the database level. This tension motivates document-to-database (Doc2DB) construction, yet most existing approaches treat it as a one-shot extraction problem, focusing on individual tables while failing to reconcile extraction results with relational semantics. We present DataMosaic, a Doc2DB system that explicitly mediates between extraction and database semantics. Given an ER schema with integrity and business constraints, DataMosaic coordinates entity and relationship extraction with verification, repair, and targeted re-extraction in a closed extract-verify-iterate loop managed by an orchestrator. By separating intrinsic document ambiguity, correctable extraction errors, and structural constraint violations, DataMosaic incrementally constructs database instances that are accurate and semantically consistent. The system supports pluggable third-party LLM extractors, domain-specific verifiers, and repair operators. Experiments across financial, legal, academic, and student datasets show that DataMosaic substantially reduces constraint violations and improves database-level accuracy over strong Doc2DB baselines.