The phrase "constant gaze" is correct and usable in written English.
It can be used to describe someone who is looking steadily or intently at something or someone without interruption.
Example: "She felt uncomfortable under his constant gaze, as if he was scrutinizing her every move."
Alternatives: "unwavering stare" or "steady look".
The world's constant gaze was clearly wearying for Lennon, but that was O.K. because he didn't always take himself seriously.
Our Peruvian journey started almost three weeks earlier in Arequipa, a town high in the Andes under the constant gaze of the El Misti volcano.
Much like the intense scrutiny of audiences' eyeballs, the camera's constant gaze doesn't seem to allow the troupe members to relax.
Thanks to the almighty smartphone, offenders can be under the constant gaze of case managers, who will monitor their activities in real time.
The asteroid Eros, now under the constant gaze of a satellite from Earth, appears to be both composed of dense material and battered by many ancient collisions, scientists reported today.
The constant gaze on my body, all the way until the day I stopped looking like a girl.
In Delhi, they identified one hotspot by a tea stall: "Men would loiter and intimidate women with their constant gazing.
Furthermore, the constant gazing makes you feel as if you're being watched, providing a confrontational experience.
More important, despite the near-constant gaze of spy satellites, U.S. intelligence agencies were stunned to learn from Israeli officials in 2007 that North Korean scientists had helped Syria build a secret nuclear reactor in the desert.
The drugs and the alcohol, then, were perhaps a way of attempting to deal with the pressure of living a life in the constant public gaze.
As women, they're subject to a constant, unforgiving gaze that has long critiqued the femininity of black women based on body type, complexion, hair and physical proximity to the European ideal.
