TOKYO—Japan is scrambling to contain its worst spill of contaminated water from the crippled Fukushima Daiichi nuclear plant since its meltdown more than two years ago, drawing fresh scrutiny to what experts say remains its shortsighted handling of the site.

On Wednesday, Japan's nuclear watchdog declared that the plant had suffered a "serious incident"—level "3" on an international scale—after operator Tokyo Electric Power Co. said that some 300 metric tons, or 79,000 gallons, of highly radioactive water...