There is a sense in which your suggestion is correct.
The gravitational field of the Earth (above its surface) is described by the Schwarzschild metric. There are several different ways of writing this down, and the one relevant to this question is called the river model - the link is to a scientific paper, but it's quite readable even if you ignore all the equations.
Written this way the position of a freely falling object is stationary, but the spacetime around it is flowing inwards towards the Earth. It's called the river model because of the analogy with being swept along by the water in a river. You may be stationary with respect to the water around you, but you're moving with respect to an observer on the bank.
Don't take this description too seriously because spacetime is not a fluid like water, and what's flowing inwards towards the Earth is the coordinate system not anything material. The point this description of Earth's gravity makes is that acceleration and gravity really are the same thing. When you're standing on Earth's surface you are accelerating outwards (at 1g), but you're accelerating relative to a freely falling observer.
Response to comment:
integralanomaly5 asks:
also constant acceleration should reach ever higher speed.. yet some how we don't reach warp speed sitting here
The correspondance between acceleration and gravity is only local i.e. it applies only within an infinitesimal region of the point you're measuring the acceleration. Having said this there is a sense in which we do reach warp speed sitting here. But bear in mind there is a degree of playfulness in the following interpretation of your question.
First, note that constant acceleration doesn't mean an object ends up at warp speed, it means the object asymptotically approaches the speed of light. For more on this see the question Does the pilot of a rocket ship experience an asymptotic approach to the speed of light?. So your question should really be:
also constant acceleration should asymptotically approach the speed of light.. yet some how we don't asymptotically approach the speed of light sitting here
Ask yourself relative to whom are we measuring our acceleration. Well, we're measuring it relative to the freely falling observer passing us. So if we're going to measure the change in velocity the acceleration on us is causing we need to measure it relative to the freely falling observer. If we are really accelerating then the velocity of the freely falling observer relative to us should asymptotically approach the speed of light.
The trouble with doing this measurement on Earth is that the freely falling observer just hits the ground with a splat. We need to take the ground away by compressing all the Earth's mass into a point at the centre i.e. turn the Earth into a black hole with use hovering at a distance on one Earth radius from the centre. Now what happens? Well it depends who you ask. From our perpsective the freely falling observer initially accelerates inwards, but as they approach the black hole their velocity decreases and actually tends to zero at the event horizon. They would take an infinite time to reach the horizon.
But in general relativity measurements are only really relevant if you do them locally. So what we need is someone else hovering just above the event horizon and ask them what speed the freely falling observer is falling. As it happens I've just answered this in response to the question Black holes and Time Dilation at the horizon. A person hovering at a distance $r$ from the event horizon sees a falling object pass at a speed given by:
$$ v = c \left( \frac{2M}{r} \right)^{1/2} $$
and at the horizon $r = 2M$ so at the event horizon a hovering observer sees the object pass at the speed of light.
Gosh I've ranted on a bit, so lets do a brief recap.
with constant acceleration the object asymptotically approaches the speed of light and reaches it after infinite time and at an infinite distance
if we're stationary at the Earth's radius from a black hole with the mass of the Earth then relative to us a falling object asymptotically approaches the speed of light and reaches it after infinite time at the event horizon
So the two situations really are analogous, though you might be justified in claiming it's a rather tenuous analogy :-)
