Earth rotates about its axis and also revolves around the Sun at the same time. So why Earth is considered as an inertial frame in Newtonian Physics. So technically, I'm effectively asking why the Earth-centered, Earth-fixed (ECEF) frame is considered an inertial frame?
You are right that it should not be considered an inertial frame for many types of problems. This is how you end up with fictitious forces to account for (such as the Coriolis effect). However, this only has practical effect for larger scale problems. For the types of problems generally considered in physics class, the inertial frame approximation will work fine. One way to look at it is that in the rotating earth case, the acceleration that you feel from circular motion $v^2/R$ will be much less than that of gravity, so you can ignore it. So for blocks falling from small heights, it won't matter very much. As the problem scales up, you'll need to take these effects into account.
To expand slightly on John Rennie's comment, almost everyone who discusses ECEF also discusses ECI, the "Earth-centered inertial" frame, and talks about how ECEF is not "inertial," in contrast to ECI. I don't know anyone who considers it "inertial" in all cases. Especially if you're dealing with weather and atmospheric physics, you have the Sun heating up air on the equator into an updraft, but this gets transformed by the Coriolis effect into a wind drifting westward relative to the surface: in a more folksy explanation, the inertial tangent frame travels with speed $r \omega$ as $r$ increases; so something moving with speed $v = r \omega$ that rises by a height $h$ into a tangent frame moving with speed $(r + h) \omega$ will appear to be moving backward with speed $h \omega.$ These west-moving equatorial winds are called the "trade winds" and the hot air that rises from that current tends to "fall" in a corresponding "east" wind at the "horse latitudes", in a "tube" of convection known as the "Hadley cell". (Confusingly these winds are called "westerlies" because ships used to mark which way the wind was blowing based on the direction it is coming from.)
All of that is Coriolis stuff; it depends on the Earth being a rotating reference frame, not an inertial one.
If someone is treating the ECEF frame as "inertial" it's perhaps legitimate if they're travelling northward near the equator (no Coriolis force; centrifugal force can be absorbed into gravitational acceleration). But in general ECEF and ECI are used by people talking about satellite navigation, and on those scales the Coriolis force usually peeks its ugly head in. The only thing I could think that would it negligible is if your satellite is orbiting the Earth many times per day, but the GPS satellites, for example, only orbit twice a day and therefore can't neglect such effects (and should use ECI to correct for them).