Physical design watermarking on contemporary integrated circuit (IC) layout encodes signatures without considering the dense connections and design constraints, which could lead to performance degradation on the watermarked products. This paper presents ICMarks, a quality-preserving and robust watermarking framework for modern IC physical design. ICMarks embeds unique watermark signatures during the physical design's placement stage, thereby authenticating the IC layout ownership. ICMarks's novelty lies in (i) strategically identifying a region of cells to watermark with minimal impact on the layout performance and (ii) a two-level watermarking framework for augmented robustness toward potential removal and forging attacks. Extensive evaluations on benchmarks of different design objectives and sizes validate that ICMarks incurs no wirelength and timing metrics degradation, while successfully proving ownership. Furthermore, we demonstrate ICMarks is robust against two major watermarking attack categories, namely, watermark removal and forging attacks; even if the adversaries have prior knowledge of the watermarking schemes, the signatures cannot be removed without significantly undermining the layout quality.