No, you cannot directly use a Mac as a TV/monitor for an Xbox 360.
Why It Doesn't Work
Video Input Missing: Xbox 360 outputs via HDMI, Component, or Composite. Macs (even those with HDMI ports) have HDMI output only—they cannot accept a video signal.
OS Limitation: macOS has no built-in support for capturing and displaying live external video from a console.
Practical Alternatives
Use a TV or Monitor: The intended, simplest solution. Any HDMI/Component display works.
Capture Card + Software: Connect Xbox 360 → Capture Card (Elgato, AVerMedia) → Mac via USB → View in OBS/Streamlabs. Adds latency; not ideal for competitive play.
Remote Play (Not Native): No official Xbox 360 remote play to Mac. Xbox One/Series support this via the Xbox app.
Avoid third-party "Mac as monitor" apps—they're unreliable, often scams, and pose security risks.
