Gravity is considered to be the curvature of spacetime by Einstein and a force by Newton. So are both notions equally correct or Einstein's perspective completely replaced Newton's thinking?
Einstein's relativity is more correct than Newton's equations for gravity. There are details to keep in mind, though.
For weak sources of gravity, like Earth, General Relativity and Newton's equations give indistinguishable answers. NASA scientists used Newton's equations to send astronauts to the moon and probes beyond our solar system because the math is easier and the answers the equations give are different by an immeasurable amounts.
However, as gravity gets stronger, like when you get close to a star, General Relativity and Newtonian equations start to give different answers. At the beginning of the 20th century, Newton's equations were failing to predict the motion of the planet Mercury. Einstein's relativity equations make correct predictions.
This often happens in science. Newton's equations are not wrong. It's better to say that we have found limits to where they give correct results. General Relativity has a larger area where it gives answers that agree with our measurements. At the moment, we have not found limits where General Relativity stops giving correct answers.
Newton's Theory of Gravity, just stated it as a force that exists, and explained how the force works but it never interpreted the origin of Gravity. He himself stated in his paper of Philosophia Naturalis Principa Mathematica that he does not know the origin of the gravitational force and leaved the task to the reader.
Einstein's General Theory of Relativity explained the origins of the Gravitational Force along with its precise working. The General Theory of Relativity is considered to be more accurate and correct and it has replaced Newton's theory of gravity because of the following reasons :-
1.) Newtons's Theory of Gravity basically explained the nature of the gravitational force but nevertheless stated its origin.
2.) Newton's Theory of Gravity was not able to explain the precession of the perihelion of Mercury, that is the periodic shift in the perihelion (lowest point of an orbit) of Mercury.
3.) According to Newton's perspective, the force of gravity acting between two bodies is directly proportional to the product of their masses and inversely proportional to the distance between them squared. So according to his theory, since light particles - photons are massless, they would not be affected by a gravitational field. But General Relativity proved that the path of a light ray is influenced by a Gravitational Field, this phenomenon is known as - Gravitational Lensing.
4.) All the Newton's theories are based on absolute space and time, whereas both the Special and General Theory of Relativity are based on relative space and time.
Due to these reasons, General Relativity is used for making precise and accurate calculations. However, Newton's Theory of Gravity is still used in some cases for weak sources of gravity like Earth because of its simplicity.