Necessary? (Insert Rip Torn from Dodgeball here)
There are no fish I can think of that will survive in a 10 gallon tank that will not thrive with the correct plants. So I would say that yes, in your situation, plants are necessary to the fish's health. In larger tanks there are fish that will uproot and/or eat plants, but there's nothing like that you will be able to house in a 10.
You can get by without ecocomplete. Exactly what you can use in place of it depends on what fish you're planning on keeping.
If you're keeping dwarf shell dwelling cichlids, livebearers, cherry red shrimp, or dwarf puffers, spring for a small box of laterite and mix it with rinsed sand (all purpose sand from home despot will work fine so long as it doesn't smell funny, pool filter sand is ideal, and I really like black sand. 3-M color quartz for mixing with pool paint is excellent) and some crushed coral or seashell (oyster shells work fine, and you can crush your own in a burlap bag with a hammer), put that in the tank first, then top with an equal amount of the sand you used in the mix. This will make the water nice and hard and give your plants some iron for their roots.
If you're keeping white clouds, skip the oyster shells, and mix in some areas of good sized gravel or pebbles (like 1/2" to 2" ). Either wrap some of the pebbles in willow moss or simply don't plant those areas. This will give any eggs a decent chance of hatching and the fry a place to hide while they're really tiny.
If you're keeping tetras, a betta, croaking/sparking/honey gouramis, cories, ottos, kuhlis, killies, or badis badis, start with a layer of peat - the chopped, compressed stuff they sell for a couple of bucks a cubic foot - about a quarter of an inch - then half an inch of organic garden soil (the stuff for growing bonsais is ideal, but for that money you might as well get ecocomplete), then top with a dark sand or small gravel. (you could add laterite or not as you like.)
Franky, in a 10 gallon tank, you could get a package of root tablets, crush about 5 of them, and mix them into the bottom third of your substrate and that would work just fine.
Make sure you match our plants to the light you'll be putting over it. Start out with some fast growing stems (which you will have to trim every week until your smaller, slower plants get going. Then you can remove the stems. If there's an aquatic gardening club in your area, you can probably pick something up on the cheap that will work ideally.) You may want to keep it to Java Moss, Java Fern, and some small cryptocorynes if you've got the standard 18 watt strip light. Not to worry - they'll look fine. Or you can get a glass lid and two strips and grow just about anything. Vallisnera works really well for hardwater species, but get the corkscrew variety - straight and narrow will outgrow a 10 in a matter of days.
You're getting yourself in for lots of fun. Check out this page
, the 2008 AGA aquascaping contestants. 10 gallons is 38 litres, so look at those tanks. You'll get an idea of what you can do with it.