WASHINGTON—The Senate on Thursday approved the most sweeping expansion of federal hate crimes law since Congress responded four decades ago to the assassination of Martin Luther King Jr.

The legislation, backed by President Barack Obama, would extend federal protections granted under the 1968 hate crimes law to cover those physically attacked because of their gender, sexual orientation, gender identity or disability.

“This bill simply recognizes that there is a difference between assaulting someone to steal his money, or doing so because he is gay, or disabled, or Latin...

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