At your speed (around 434 km/s), a mote of dust weighing a single milligram will have an impact with the same energy as a 20-gram TNT explosive, so unless your ship is capable of tanking hit after hit after hit from military-grade missiles, it definitely won't be able to fly anywhere near the speeds you propose. At a speed of 252'000 km/h (much, much less than your proposed 1'562'500 km/h), the same mote of dust will still hit like half a kilogram of TNT, which will probably still destroy your ship.
Sounds like you need to be able to stop the damage, because otherwise, the damage will definitely stop you before you can get anywhere near as fast as you want.
The issue you refer to is the hypervelocity impact that even small micrometeoroids could have on your ship (or conversely, the hypervelocity impact that your ship would have on people). For the latter, the solution is simple: have the ship's computers prevent the orbital maneuvers required to direct a ship into a planet's surface at speed. Orbital mechanics require some thought going into them, and adding a computer to do it for you has the benefits of a) the pilot not having to do those calculations and b) the ship being able to prevent the use of a starship as a kinetic kill vehicle.
Would a computer still be able to be hijacked and used as a weapon? Of course, but why even use a ship? If you want to destroy something at great distance in the exact same way but for cheaper and without expending human lives, simply attach one of your engines to a giant fuel tank and fire it straight towards your enemy. Whatever engine works for your spaceship will just as well work for kinetic kill vehicles.
As far as preventing damage to your ship, you may be interested in the Whipple shield. This is a real technology that is in use on the ISS that uses spaced armor to disperse the energy of the impact to make it more manageable before the projectile hits the ship. This consists of several relatively-thin plates, not intended to stop or even slow down the micrometeorite that much, but rather to break it up into smaller pieces which individually carry less energy and can be more effectively deflected by regular armor.
In your case, a simple plate of armor won't do - as other answers point out, a micrometeorite encountered by a ship moving at your proposed speeds would have an energy similar to that released by a nuclear bomb. If a regular Whipple shield as displayed on Wikipedia were used, it would probably just get punched through and vaporize the whole ship.
So what if you could just deflect the danger directly instead of fragmenting it? Imagine a large graphene net suspended by poles from your ship, forming a bubble around it (or maybe just a net in front if you're a cheapskate spacefarer). The damage from a micrometeorite at those speeds would certainly not be absorbed at all by the net, but the graphene might deflect the micrometeorite slightly, and if the net is placed sufficiently far in front of the ship and shaped in the correct way (as in an ellipse with the ship between the two foci), it would substantially reduce the probability of the ship getting hit by the projectile. The projectile wouldn't be slowed down in the slightest, but it would be out of the way of your ship, and that's what matters. All you have to do is implement a system that lets you swap out the nets when you arrive at your destination, so that any holes that got punched can be filled to minimize the risk of getting hit by a micrometeorite with enough kinetic energy to destroy a tank.
Safe travels.