it's not entirely clear how do you define an planet being "Earth-like" (to me, a planet with a surface as hot as 1000K is nothing Earth-like!), but I'll try ti give a general idea what makes a planet a planet instead of a diffuse gas cloud.
A planet does not fall apart because it's a piece of solid rock. On the contrary, a pretty solid planet (or a smaller piece of rock, like the recently imaged Ultima Thule) will fly apart in pieces if it rotates too fast. The main point to take away here is it is unimportant (or, pedantically, nearly entirely unimportant) what is a celestial body made of. It can be a pile of loose sand, or a huge ball of liquid. It can be even a ball of hot plasma, just like stars are, provided they are massive enough that the matter--the hot gas--does not escape the gravity well of the object too quickly.
So, if the planet temperature is high enough that it melts as a whole, this lump of liquid lava will still be a whole normal planet, spherical if it does not rotate on its axis, or geoidal if it does; the faster is the rotation, the more flattened is the shape along the rotation axis.
So I would say that 1000°C is not even close to the point where a rocky planet starts losing mass due to evaporation, assuming it's composition is close to that of Earth.
As the temperature grows, you'll get more material in a gaseous form. The loss of mass (simply speaking, the gradual evaporation of the molten planet) requires that a significant portion of gas molecules posses enough energy to overcome gravity of the body. The effect will be dampened if the planet has atmosphere: hot gas rising and trying to escape will inevitably "bump into" colder gas of upper atmosphere (if you prefer more physical lingo, it will exchange kinetic energy with colder gas, so it's temperature evens out). Here we assume that (1) the atmosphere is transparent enough (so that it's cooler as we go farther from the surface), and the major heating of evaporating material happens at the body surface, which absorbs the most star's radiative energy, and (2) the atmosphere does exits at all, instead of being completely blown out (or significantly rarefied) by the star's stellar wind of highly energetic particles.
If these two assumptions hold, the planet can exist for millions of years as a molten ball of whatever it's made of without dissipating, and, back of the envelope check, the temperatures of a few thousand K do not seem to imply that the body becomes gravitationally unbound due to material evaporation, when gas kinetic energy overcomes the gravitational potential energy of the planet.