Though it meant simply "heavy" (gravis) in Latin, it came to English in the sixteenth century through French (gravité) where it meant 'dignity' or 'seriousness'.
It's first use in the scientific sense seems to have been through a metaphor in a poem (Psychodoia Platonica: or, a Platonicall Song of the Soul, consisting of foure severall Poems) by Henry More (1614-1687):
What they pretend of the Earths gravity,
Is nought but a long taken up conceit:
A stone that downward to the earth doth hy
Is not more heavy than dry straws that jet
Up to a ring made of black shining jeat.
Each thing doth tend to the loud-calling might
Of sympathy. So 'tis a misconceit
That deems the earth the only heavy weight.
A few years later the term was taken up in scientific treatises, but with the meaning "weight":
"...and in very truth, the greatest force of gravity doth appear in those bodies which are eminently earthy..." (Sir Kenelm Digby, 1645)
Over the next forty years (though it retained the meaning "weight" until the very end) it was used with increasing mathematical rigor, culminating with Newtons publication of the laws of motion in 1684.